DE4118302A1 - DEVICE AND METHOD FOR ANALYZING A MECHANICAL MULTI-BODY SYSTEM - Google Patents

Description

A method for effectively analyzing a ho lonom, non-holonomic or partially non-holonomic Multibody system described, which defines the General type topology of the multibody system and that Creating symbolic equations requires which for the equations of motion of the general multibody system are representative. The method further comprises the on give numerical values, which a specific multicore persystem defined in the general way that Simulate the operation of the multibody system and that Get the simulation results. The process includes checking and displaying the suitability of physical and restrictions imposed by the system regarding the nu values that have been entered.

It becomes a method for analyzing a mechanical Systems described, which interconnected me  chanic elements that defines the basic Mechanical system topology, choosing one by inertia fixed frame of reference for the mechani cal system, specifying the limit conditions of the mechanical system and the creation of symbolic movements equations for the forced system for a holonomic, non-holonomic and partly non-holonomic dynamic System from the specified system limit value conditions includes.

According to a further aspect of the invention, a device device for analyzing a mechanical system the connected mechanical elements suggested wel che a preprocessor, a data input device for Enter the topology and limit conditions of the Sy stems in the preprocessor and includes a device that with the preprocessor to formulate symbolic movement equations for a holonomic, non-holonomic and partial non-holonomic limited mechanical system on the Basis of the entered topology and limit value conditions is coupled.

The invention is based on exemplary embodiments Hand explained in more detail in the accompanying drawing. In the drawing shows

Fig. 1A is a view of an analyte me chanical multibody system

FIG. 1B, the mechanical multibody system of Fig. 1A in a dissolved state,

Fig. 2 is a block diagram of the system,

Fig. 3 is a block diagram of the function of the Vorpro zessors in the described system,

Fig. 4 is a diagram of a tree topology depicting lung from the preprocessor,

Fig. 5 is a block diagram of the functions exerted by the module system data,

Fig. 6 is a block diagram of the functions exerted by the nachgeschalte th processor,

Fig. 7 shows a further mechanical multibody system, which is to be analyzed.

The invention provides a movement and force analysis of a mechanical system of interconnected rigid bodies, such as, machines, robots, vehicles and space vehicles. The system uses the Kane method for the mulate the equations, what movements the mechani systems and the forces determine how this in "Dynamics: Theory And Applications", Thomas R. Kane and David A. Levinson, McGraw-Hill Book Company, Copyright 1985 (hereinafter referred to as Kane). The Kane Ver driving is using symbolic computer algebra techniques couples to a specific simulation description for a to create a given multibody system. Reference is genome men on the dissertation "Symbol Manipulation Computer Pro gram Generating Equations Of Motion Of Mechanical System ", Yi-Chung Chang, Ph.D. dissertation, Stanfort University Library, 1989.

A mechanical multibody system can always be solved by breaking connections between two of the bodies in the system so that the system has a "tree" or open topology. A mechanical multi-body system is shown in FIG. 1A, in which elements or bodies are connected to one another within the mechanical system at the articulation points 11 and 12 . The same system solved at the hinge points 11 and 12 is shown in Fig. 1B ge. It is possible to maintain more than one "tree" in one system. Degrees of freedom between any adjacent elements or bodies within the system can be introduced by specifying generalized coordinates and generalized velocities with respect to the relative location and movement between these two bodies. The total number of generalized coordinates is equal to that of the generalized speeds. As a result, the system is holonomic. The approaches required to simulate a holonomic multibody system are described below. When the system is released, as shown in Figure 1A, to create a "tree" topology, limit conditions can be imposed on the relative position or movement between two bodies within the system, these bodies being separated during the solving process to re-establish the connection. As a result, the generalized speeds are no longer independent and the system becomes non-holonomic. The approaches and formulations for simulating a non-holonomic multibody system are also described below. A partial multibody system can be described by imposing different sets of limit conditions on a system described as a "tree" topology, the formulations for which are also discussed below. Three types of simulation can be carried out with the described invention depending on the holonomic and / or non-holonomic degrees of freedom of the multibody system. This includes kinematic simulation, reverse kinematic simulation and dynamic simulation, as explained below. An energy test function is proposed in order to create great confidence in the numerical results obtained from these different simulations. This test function is also described in detail below. A "tree" topology multi-body system contains µ rigid bodies Bi (i = 1,..., Μ), which are defined to be connected to a predetermined body or an inertia reference N. Bo, as shown in FIG. 1B, can be used to represent the inertial reference N. Thus, N or Bo is always the root of the "tree" topology. The tree can be defined by an arrangement j (i) a dimension µ, and j (i) is the index number of the body to which the body Bi is connected. For example, the j (i) arrangement of the multibody system in Fig. 1B is as follows when the multibody elements are labeled as shown in Fig. 1B.

i 1 2 3 4 5 6 7 8
j 0 1 2 3 2 5 6 6

Bi has a mass of mi and a central inertia dyad of Ii. The configuration and movement of Bi with respect to Bj can be specified as an equal number of generalized coordinates qr and generalized velocities ur. After specifying qr and ur for all Bi, where i ranges from one to µ, there is a total number of qr and therefore ur of n. The mechanical system then has n degrees of freedom. Bi * and Bj * are the mass centers of Bi and Bj. ( N ₁ ⁱ , n ₂ ⁱ , n ₃ ⁱ ) and ( v ₁ ^j , v ₂ ^j , v ₃ ^j ) are two sets of orthogonal unit vectors, fixed in Bi and Bj, and both sentences coincide with ( n ₁, n ₂, n ₃), a set of orthogonal unit vectors, fixed in N (or Co). The above exists when all qr are zero. It should be emphasized that at j is zero Bo * is a point fixed in Bo, and the unit vectors n fixed in Bi and Bj are equal to the unit vectors fixed in N. Furthermore, if i and j are two points that coincide with each other without delay and are fixed to Bi and Bj without delay, the configuration and movement of Bi with respect to Bj is given in terms of qr and ur as follows:

P^yy is a position vector of B_j* to_j in B_j-Bases.
Pⁱⁱ is a position vector from_j till B_i* in B_i-Bases.
^jC.ⁱ is a direction cosine vector of (v₁^j,v₂^j,v₃^j) to (n₁ⁱ, n₂ⁱ, n₃ⁱ)
V^rel is the relative speed between_i and_j, whichV ⁱ-V ^j in B_i-Bases is.
^jwⁱ is the angular velocity of B_i in relation to B_j in B_i-Bases.

If j is zero, o is not necessarily delay freely fixed in N, and its speed is given by

V ^o , which is the velocity of _o in N bases or frame of reference.

Furthermore, the relationship between ur and r (the time from qr) through the matrices Yrs and Zr as follows given:

There are αi forces F ^ik (k = 1,..., Α _i ) that act at point P ^ik , with torques T ′ acting on the body Bi. These sizes are defined as follows:

F ^ik is a force expressed in B _i bases or N bases.
P ^ik is a position vector from B _i * to P ^ik .
T ⁱ is a torque expressed in B _i bases or N bases.

The kinematics include position, orientation, partial speed, Partial angular velocity, acceleration and angular acceleration leaning of each element or body and the expressions for the time derivatives of qr in expressions of ur and qr (r = 1,..., n). These sizes, which relate to a body Bi by summing the sizes for Bj and Obtain "relative" sizes between Bi and Bj, whereby the "Tree" topology order followed by the root to branches becomes.

The following relationships apply:

Direction cosine matrix from ( n ₁, n ₂, n ₃) to ( n ₁ ⁱ , n ₂ ⁱ , n ₃ ⁱ );

^o C ⁱ = C ^{j o j} · C ⁱ (1)

Position of B _i *;

P ⁱ = P ^j + ^j · P ^o C ^o C ^jj + ⁱ · P ⁱⁱ (2)

Partial angular velocity of the body Bi;

w _r ⁱ = ⁱ · C ^j _r ^j w ^j + w _r ⁱ (.. r = 1,., n) (3)

in bi-bases.

w _t ⁱ = ⁱ C ^j w _t ^j + ^j w _t ⁱ (4)

in bi-bases.

w ⁱ = ⁱ C ^j * w ^j + w ^{i j} (5)

in bi-bases.

Partial speed of B _j *;

If j ≠ 0,

V _r ⁱ = ( V _r ^j ) _N + ⁱ C ^j · [( V _r ^j ) - w _r ^j × P ^jj ] + w _r ⁱ × P ⁱⁱ + V _r ^rel (6)

( V _r ^j ) _N in N bases or Bi bases.

V _t = ( V _t ^j ) _N + C ^j · (( V _t ^j ) _L + w _t ^j × P ^jj ] + w _t - ⁱ × P ⁱⁱ + V _t ^rel (7)

V _r ^j = ( V _r ^j ) _N + ( V _r ^j ) _L , N and Bj bases respectively.

V ⁱ = V ^j + ^o C ^j · ( w ^j × P ^jj ) + ^o C ⁱ · ( w ⁱ × P ⁱⁱ + V - ^rel ) in N-bases (8)

or j = 0;

V _r ⁱ = V _r ^o + w _r ⁱ × P ⁱⁱ + V _r ^rel (9)

V _r ^o in N bases or Bi bases.

V _t ⁱ = V _t ^o + w _t ⁱ × P ⁱⁱ + V _t ^rel (10)

V ⁱ = V ^o + ^o C ⁱ · ( w ⁱ × P ⁱⁱ + V ^rel ) (11)

in N bases.

Angular acceleration of Bi;

in B _i bases, no contribution from _r .

Total angular acceleration

in B _i bases.

Acceleration of B _i * if j ≠ 0,

is time differentiation of V ^rel in B _i bases, no contribution from _r , time derivative from u _r .

Total acceleration,

q _r in terms of u _r and q _r , using

[Wrs] = [Yrs] ^-i (17)

{Xr} = [Wrs] · {Zr} (18)

and

The dynamic equations can be found in the following Format to be written:

[Mrs] · {r} = {Rr} (20)

In the above, [Mrs] is an n × n matrix, {r} is an n × 1 matrix with r, which is the time derivative of ur is and - {Rr} - is an n × 1 matrix. The kinema table sizes to which reference is made become Define M and R used.

M is the system mass matrix.

where m _{i is} the mass of the body B _i .
V _r ⁱ , V _s ⁱ from equation (6) or (9).
I ⁱ is inertia dyad of body B _i , there

I ⁱ = I₁₁ ^L n ₁ ⁱ n ₁ ⁱ + I₂₂ ^L n ₂ ⁱ n ₂ ⁱ + I₃₃ ⁱ n ₃ ⁱ n ₃ ⁱ + I₂₃ ⁱ ( n ₂ ⁱ n ₃ ⁱ + n ₃ ⁱ n ₂ ⁱ ) + I₃₁ ⁱ ( n ₃ ⁱ n ₁ ⁱ + n ₁ ⁱ n -₃ ⁱ ) + I₁₂ ⁱ ( n ₁ ⁱ n ₂ ⁱ + n ₂ ⁱ n ₁ ⁱ ) (22)

w _r ⁱ , w _s ⁱ from equation (3)
R is the right side of the dynamic equations. First find the resulting force F _i and resulting moment T _i around the center of mass of all forces acting on each body B _i , whereby

in N bases or B _i bases.

Then construct inertia torque, no contribution from r, every body, bi

T _i * = I ⁱ · α ⁱ + w ⁱ × (I ⁱ × w ⁱ⁾ in B _i -Basen (25)

I ⁱ is inertia dyadie of Bi, equation (22)
α ⁱ is from equation (12)
w ⁱ is from equation (5)
Get Rr by:

V _r ⁱ from equation (6) or (9)
m _i mass of body B _i
ª ⁱ from equation (4) or (16)
F _i from equation (23)
w _r ⁱ from equation (3)
T _i from equation (24)
T _i * from equation (25)

The constraining force on body Bi means the resulting force and moment around a point that is either fixed in Bi or Bj (the body to which Bi is connected by the "tree" topology), which is added to Bi by Bj to the given active forces and torques must be exerted in order to create the movement solutions for the system. The problem is as follows. If the movements of the system, all active forces and torques, a body Bÿ and a point P _c ⁱ , are fixed either to Bi or Bj, expressed in terms of a position vector Pc from Bi * or Bj * to the point in question the body-fixed base are all given, F _c ⁱ , the force acting on Bi by Bj in these bases or references and T _c ⁱ , the force that must act on Bi via Bj in these bases or references must be determined. To solve this problem, who took the following procedures:

• 1. Find all bodies including bi one the branches of bi are. Then it's a Ge total number of NTR bodies available, the Zif remotely stored in the ITR (k) series, where k = 1,. . . , NTR is.  
• 2nd in N bases, where index ITR (k) applies to m, ª _T and F.
  m _{ITR (k)} is mass of body B _{ITR (k)} .
  ª _T is obtained from equation (15)
  F _{ITR (k)} from equation (23), dissolved in N bases.
• 3. If P _c ^{i is} fixed to B _i , then ^o P _c ^L ; Location of P _c ⁱ is ^o P _c ⁱ = P ⁱ + ^o C ⁱ · P _c (28) in N bases, where P ^{i is} from equation (2) and
  ^o C ⁱ from equation (1), or if P _v ^{i is} fixed to Bj, then ^o P _c ^L = P ^j + ^o C ^j · P _c (29)
• 4. T _T * for all bodies, specified in ITR (k) series ( T _T *) ⁱ = I ⁱ α _T ⁱ + w ⁱ × (I ⁱ · w ⁱ ) (31) in bi-bases, where
  T ^{i is} the inertia dyad from equation (24).
  α _T ⁱ is from equation (13).
  w ⁱ is from equation (5).
• 5. in N bases, where index ITR (k) applies to
  T _T *, T , P, m, ª _T and F ( T _T *) ^{ITR (k)} is from equation (31).
  T _{ITR (k)} is from equation (24).
  P ^{ITR (k)} is from equation (2).

If a "tree" topology multibody system is defined, limit conditions can be imposed on the system,  a multibody system with either configuration or To generate motion limit conditions. Configuration and motion limit conditions become as follows Are defined. It should be emphasized that these two types from limit value conditions to limit value conditions the relative configuration or movement between two Relate bodies. A general type of limit value condition or constraint that can be used to Limiting system movements is discussed below. A configuration limit condition is given by the fol defined sizes.

Two bodies Bi and Bj

Two position vectors, which appear as P _i ^G , P _k ^H in Bi bases and Bk bases, respectively. P _i ^G is the position vector from B _i * to a point G, and P _k ^H is the position vector from B _k * to a point H ^k . These two vectors are functions of qr and t or constants. A unit vector b , fixed in B _i or B _k , expressed in B _i or B _k bases.
Constant:
A timing is defined as

where f _T is or can be a constant; ^N P ^G and ^N P ^H are positions of G and H in N.

A motion limit condition can be by the following Sizes are defined:

Two bodies B _i and B _k

Two position vectors P ^G , P ^H · P ^G is from B _i * to a point G and can be broken down into two components P _i ^G and P _ii ^G , each expressed in B _i bases and B _ii ⁱ bases.

B _ii is a body in the system other than B _i .

P ^G = P _i ^G + P _ii ^G (34)

similar

P ^H = P _k ^H + P _kk ^H (35)

A unit vector b in selected body bases, ie B₆ bases. Constant components.

A time function f _T is defined as:

where V ^G and V ^{H are} velocity corresponding to points G and H, assuming that they are fixed instantaneously in B _i and corresponding to B _k .

A general limit value condition or limit value can be given by Creation of coefficients Ar (r = 1,..., N) and B as Functions of time and qr of subsequent non-holonomic ones Limit equation can be specified.

If all the limit value conditions mentioned are imposed, the the system has p Degrees of freedom, where

p = n-m (38)

is.

The following kinematic derivation is for the purpose creating an equation of limit condition, as given by equation (37) for each of the configuration  or movement limit conditions are given, and by Differentiations to the assigned limit equation in r to create sizes D like

to produce, where D can be expressed as

By combining these equations for all limit conditions can measure the entire equations in one measure trixformat can be described.

[Aer] {U _r } + {Be} = 0 (41)

[Aer] { _r } + {De} = 0 (42)

In equations (41) and (42) [Aer] is an m × n matrix, {ur} and {r} are n × 1 matrices containing ur and r and [Be] and [De] m × 1 matrices. The configura tion limit conditions are in terms of qr and t expressed explicitly to determine a set of to facilitate qr and t which satisfy these equations.

^N P ^G = P ⁱ + ^o C ⁱ · P _i ^G (43)

^N P P ^H = ^{k o} + C ^k · P _k ^H (44)

in which
P ⁱ , P ^k from equation (2),
^o C ⁱ , ^o C ^k from equation (1).

Then apply equation (3) to set configuration limits value condition equations in expressions of qr and t to build,

g (qr, t) = 0 (45)

Then get assigned motion limit condition Equation in the form of equations (37) and (39)

in N bases or B _i bases.

V _r ⁱ from equation (6) or (9).
w ₁ ⁱ from equation (3)

Time derivative of P _i ^G in B _i reference in B _i bases. Equation (19), etc.

in N bases or B _i bases.

V _t ⁱ from equation (7)
w _t ⁱ from equation (4)

V ⁱ in N bases or Bi bases from equation (8) or (11)

ª ⁱ in N bases or Bi bases
ª ⁱ from equation (14) or (16)
α ⁱ from equation (12)
w ⁱ from equation (5), then likewise

Let b = i, if b is fixed in B _i and b = k, if b is fixed in B _k , then:

Ar = b · [ V _r ^G - V _r ^H - w _r ^b × ( ^N P ^G - ^N P ^H ) (54)
V _r ^G from equation (46)
V _r ^H from equation (47)
w _r ^b from equation (3)
^N P ^G , ^N P ^H from equations (43) and (44).

These are the compulsory equations in the form of the equations (37) and (39).

The following describes how a set of qr and t is obtained, which all configuration limits value conditions satisfied. If L is the total number of these Limit value conditions, all configuration Threshold conditions can be combined to get:

ge (qr, t) = 0 (e = 1,..., L) (57)

[Aer] {Ur} + Be = 0 (r = 1,..., N) (58)

To use the Newton-Raphson method to solve the nonlinear set of equations of equations (57) demonstrate, assume that the first L qr Unbe are known, whose values have to be solved and the rest of the qr have fixed known sizes. The following matrix will receive:  

first L pillars, where [Yrs] is an n × n matrix that is in the relationship is defined for the solution for ur. With the equations (57), (59) and an initial estimate of the value of qr (r = 1,..., n) is the solution for qr with the Newton-Raph son method possible.

Deriving the constraints on movement (motion con straint equations) follows.

V _r ^G = V _r ⁱ + w _r ⁱ × P ^G (60)
in N-bases or Bi-bases
V _r ⁱ is from equation (6) or (9)
w _r ⁱ is from equation (3)
P ^G is from equation (2) in bi-bases and
V _t ^G = V _t ⁱ + w _t ⁱ × P ^G (61)
V _r ^H = V _r ^k + w _r ^k × P ^H (62)
V _t ^H = V _t ^k + w _t ^k × P ^H (63)
V ^G = V ⁱ + w ⁱ × P ^G , (64)
V ⁱ in N bases or Bi bases.
V ⁱ from equation (8)
w ⁱ from equation (5) and
V ^H = V ^k + w ^k × P ^H (65)

ª _i in N bases or Bi bases

assuming that G is currently fixed in Bi, in N
ª ⁱ is from equation (14) or (16)
w ⁱ , w ⁱⁱ are from equation (5)
P _i ^G , P _ii ^G from equation (34)

in bi-bases

in B _ii bases
α ⁱ from equation (12) and

Ar = b · [ V _r ^G - V _r ^H ] (68)

b from equation (36)
V _r ^G from equation (60)
V _r ^H from equation (62)
B = b · [ V _t ^G - V _t ^H ] - f _T (69)

ª ^G from equations (66) and (67).
w ^b from equation (5).
V ^G , V ^H from equations (64) and (65).

Since Ar and B in this case are given as functions of qr and t ben is in the case of the general limit condition only necessary to determine D using the relationship:

in which

The dynamic equations are the combination of the holo nomen dynamic equations, those for the "tree" - Topology system before imposing the limit conditions with the addition of the contribution from the Lagrange multiplier λe, assigned to each limit condition and the Ge totality of the limit value equations. This equation  gen can from equation (20) with the contribution of λe and equations (41) and (44) are written as:

[M] {} = {R} + [A] ^T {λ} (74)

[A] {U} + {B} = 0 (75)

[A] {} + {D} = 0 (76)

In equations (74), (75) and (76), M is an n × n matrix, R an n × 1 matrix, lambda an m × 1 matrix, A one m × n matrix and B and D are m × 1 matrices. It should still be be emphasized that n is the holonomic degrees of freedom of the mechanical system and m is the total number of System limit value conditions. The non-holonomic Degrees of freedom p are therefore p = n-m, as in Glei chung (38) is shown.

To simplify the discussions, assume that the first pur, referred to as uI, as an independent generalization nert speeds are selected, with the lead are described with uD and dependent generalizations mean speeds are. The matrix A can do this follow to be broken down into two parts AI and AD, and the glide (75) and (76) can be written as follows the:

Similarly, M and R can be decomposed, and the Equation (74) can be written as follows.

The following explanation of the problem can be made the. If M, R, A, B, D and uI are given, uD, I (to be integrated), D (only for calculating acceleration tendencies to use) and λ can be found.

The following discussion of the solution to this mathematical Problems is divided into two categories. The first Kate gory deals with a dynamic case, i. H. p is not equals zero. The second category deals with a kinemati case in which p is zero. In the dynamic In the case where p is not zero, the Ma trices for Ac, Bc and Dc constructed as follows:

Ac = -A _D ^-1A _I (80)

Bc = -A _D ^-1B (81)

Dc = -A _D ^-1D (82)

Ac is an m × p matrix and Bc and Dc are m × 1 matrices. Then get uD from the following relationship:

{U _D = [Ac] {U _I } + {B _i } (83)

Construct the matrices for Mc (p × p) and Rc (p × 1) as follows:

M _C = M _I + M _{IDA C} + (M _{IDA C} ) ^T + A _C ^T M _D A _C (84)

R _C = R _I + A _C ^T _D - (M _ID + A _C ^T M _D ) D _C (85)

Then I get about the relationship:

{ _I } = [M _C ] ^-1 {R _C } (86)

Get D from the relationship:

{ _D } = [A _C ] { _I } + {D _C } (87)

Get λ from the relationship:

{λ} = A _D ^-T [-R _D + (M _ID ^T + M _D A _C ) _I + M _D · D _C ] (88)

It is emphasized that equation (86) dynami equations of the non-holonomic system.

In the kinematic category, where p is zero A is an n × n matrix and all ur are dependent Variable. u and are obtained through the relationships:

{U} = -A ^-1 · B (89)

{} = -A ^-1 · D (90)

λ is obtained from the following relationship:

λ = A ^-T · [-R + M ·] (91)

All r are not integrated, but only used, to get acceleration information.  

The constraints for a non-holonomic system exist from two types of forces or torques. The first Kind includes the force that is required to overcome any coercion to maintain the condition described "Tree" topology system is imposed. These powers are on the Lagrangian multipliers. The second type of Coercive force is the coercive force or torque that two adjacent bodies in the "tree" topology stands. However, these forces and moments have to be modified to be the contribution of the first type of coercive forces or include moments. With a force from the Constraint equations are not an additional calculation required. It's just a fact of how the La grange multiplier is interpreted as it from the Equations (88) or (91) as interaction forces between the two bodies is obtained, their relative movement is inevitable due to the limit value equations. This physical interpretation is only for the Kon configuration and movement constraints, but not possible for a general limit value condition.

For a configuration limit value condition, the Lagrangian multiplier λ can be interpreted as follows: If b is fixed in Bi, λ is the measurement number of the force acting on Bi by Bk at point H in direction b . Alternatively, if b is fixed in Bk, λ is the measurement number of the force acting on Bi through Bk at point G in the b direction. For a motion limit condition, the Lagrangian multiplier is interpreted as the measurement number of the force acting on Bi over Bk at point G in the b direction, and minus λ is interpreted as the measurement number of the force acting on Bk over Bi im Point G in the b direction acts.

The constraint approaches are valid except that Fi and Ti must be modified before applying equations (30) and (32). This modification consists of the Lagrangian multiplier λ for the purpose of considering the contribution. As mentioned above, an evaluation of the constraining force between Bi and Bj is not possible if the multibody system to be analyzed contains general constraints. Each configuration limit condition must therefore be modified as follows:

If b is fixed in Bi:

F _i ← F _i + λ b (92)

F _k ← F _k - λ b (93)

T _i ← T _i + ( ^N P ^H - P ⁱ ) × (λ b ) (94)

T _k ← T _k + P _k ^H × (-λ b ) (95)

when b is fixed in B _k ;

F _i ← F _i + λ b (96)

F _k ← F _k - λ b (97)

T _i ← T _i + P _i ^G × (λ b ) (98)

T _k ← T _k + ( ^N P ^G - P ^k ) × (-λ b ) (99)

There is a modification for each movement constraint required as follows:

F _i ← F _i + λ b (100)

F _k ← F _k - λ b (101)

T _i ← T _i + P ^G × (λ b ) (102)

T _k ← T _k + P ^H × (-λ b ) (103)

Three ways of imposing limit conditions a "tree" topology mechanical multibody system  introduced to a simple non-holonomic multibody system. A partial multi-body system can can be defined by imposing a part or an "Sentence" from the total limit conditions on the "Tree" topology system at a time so that np distinguished non-holonomic systems. This one non-holonomic systems can be created by differentiating sets of limit condition equations to the same che "tree" topology system are imposed, the Be drawing "partial multibody system" used to To designate collection of these different systems. This collection of systems can then be in a single Simu program for analysis. If Kinematics and dynamics already for the "tree" topology more body system are derived and Ar, B or D for each Limit condition are constructed (configuration, ev supply or in general) it is only necessary to have one Collect the set of Ar, B and D, according to which limit conditions of value are to be imposed and the matrices {Aer}, {Be} and {De} to form as in Equations (41) and (42) are defined for the non-holonomic time part. The Equations (74) to (91) can then apply to any non-holo nome part can be applied.

To simulate a mechanical partial multibody systems are two additional pieces of information required. The first is how the analysis system can determine which part at the current time is to be simulated. This involves determining the initially part to be used and the criteria for switching one part to another. The second required informa tion is what the analytical system should do when the configuration or movement of the Systems just before the transition did not change the configuration or Movement limit conditions after switching to  new part fulfilled for simulation. In the simulation system, these two pieces of information are provided by the model mechanical system.

The energy test function is carried out as follows:

Gr, Qr, S,, K₂ and K₀ are defined as

V _r ⁱ from equation (6) or (9)
F _i from equation (23)
w _r ⁱ from equation (3)
T _i from equation (24)

Asr from equation (41)
λ from equation (88) or (91)

m _i is mass of body B
V ⁱ from equation (8) or (11)

w ⁱ from equation (5).
I _i is inertia dyadic of B _i from equation (22)

K₂ = ½ {U _r } ^T · [M] · {Ur} (108)
U _r is generalized speed
M is a holonomic mass matrix, equation (21)

m _i is the mass of the body B _i
V _t ⁱ from equation (7) or (10)
I _i is inertia dyad of B _i
w _t ⁱ from equation (14).

The energy test function E is defined as:

E = Z + K2 - K0 (110)

In equation (110), Z is the integration of after Glei chung (107). The integration should be from the integration accompanied by r (r = 1,.., n-m) and r (r = 1,..., n) be. If E does not remain constant, the simulations are results not exactly. If E remains constant, guaranteed this is not the validity of the simulation, but is as Accuracy indicator.

The kinematic simulation shows that no general meant coordinates qr or generalized speed  only when defining the "tree" topology mecha African multibody systems have been introduced. With an whose words is n = zero. In this case, only  in equation (107) in the computer energy test function E integrated. It should be emphasized that the sizes in Glei chung (108) and the summation of

in equation (107) are zero. In addition, since V ⁱ = V _t ⁱ , the energy test function E in equation (110) becomes

An inverse kinematic simulation indicates that the total number of qr introduced into the "tree" topology mechanical system is equal to the total number of limit conditions. In other words, n = m. The system therefore has zero degrees of freedom, ie p = zero. In such a case, only r (r = 1,..., N) from equation (16) and from equation (107) are integrated. ur and r (r = 1,..., n) are obtained by using equations (89) and (90) accordingly, and λr is obtained via equation (91). Body movements are obtained by the equations (1), (2), (5), (8), (11), (13) and (15) at each time stage, and the quantities F i and T i (i = 1 ,..., u) are calculated using equations (23) and (24) and (92) to (103) to facilitate the evaluation of the constraining force, as explained.

Dynamic simulation of a mechanical multi-body system means that after imposing constraints, the multi-body persystem has non-zero degrees of freedom, d. H. p = n - m greater than zero. The dynamic simulation can be divided into two categories gories are divided: (1) "tree" topology system (m = 0); and (2) multi-body system with limit value conditions (m not zero).  

For both cases, r shown in equation (19) and that shown in equation (107) will be integrated. Body movements are calculated using equations (1), (2), (5), (8), (11), (13) and (18) at each time step. In addition to these calculations, the following quantities will also be calculated. In the case of a "tree" topology system (m = zero), equation (20) will be solved for r, and r will be integrated to get ur (r = 1,..., N). F i and T i (i = 1,..., Μ) are calculated using equations (23) and (24) at each expression step to obtain a threshold condition estimate previously discussed. If one considers a system with limit value conditions (m not equal to zero), I from equation (86) is integrated in order to obtain uI. UD from equation (83), D from equation (87) and λ from equation (88) will also be estimated at each expression step. In addition, F i and T i (i = 1,..., Μ) will be calculated using equations (23), (24) and equations (92) through (103) to enable a threshold condition estimate discussed hereinbefore .

As can be seen in FIG. 2, this system begins with information input by a modeler defined as an individual with a relatively high level of training in characteristics and analysis of mechanical systems. The inputs are made through a keyboard or other input device 14 to a preprocessor 13 . The output of the preprocessor is connected to a symbolic generator 15 , which functions as a converter for generating new information. Certain files are generated by the symbolic generator and appear as simulation source code and documentation 16 . The output from module 16 of FIG. 2 is transferred to a text editor and compiler 17 through media such as floppy disks, tapes, and printouts. Other user inputs that can be entered through the same or a different keyboard 14 can be made by the modeler or by a user familiar with the particular mechanical systems being analyzed, but not necessarily as well the mechanical systems understands like the modeler. It should be noted that the media output is included and portable by the simulation source code module 16 and can be used by a user or a modeler on site or remotely as desired and convenient. The output from the text editor and compiler 17 is coupled to a system data module 18 , along with the user's input via the keyboard 14 . The system data module records symbols, initial conditions and time parameters for a simulation. Files generated by the system data module are supplied as an input to the simulator 19 . The files are provided to the simulation execution code 20 to create the current mechanical multibody simulation. The results of the simulation are forwarded to a postprocessor 21 , which is coupled to the user input keyboard 14 and to the system data module 18 . The postprocessor enables a user to manipulate simulation result data to produce files containing selected amounts in a custom format at 22 , xy coordinate data at 23, and motion picture simulation results at 24 . The files, coordinates and moving images 22 , 23 and 24 can each be displayed according to a user's requirements. When applying the invention, the C language is used for input to the symbolic generator 15 and the Fortran language is used for output for practical reasons. Other languages can be used, although the type described is currently the best known.

Some expressions should be defined for better understanding. A symbol is defined as a character string, which in this exemplary embodiment mostly contains 20 characters starting with a character of the alphabet or a numerical value. Any chain starting with T is a chain reserved for a symbol. A time function is a function of time only, and the name of a time function begins with the characters TT, followed mostly by 15 characters. A Q function is a function of time, generalized coordinates qi, and a position or orientation of the bodies. The name of a Q function begins with TQ, usually followed by 15 characters. A sensor function is a function of time, generalized coordinates qi, generalized speed ui, a position or orientation of the bodies, a speed or of angular velocities of the bodies or other defined sensor variables. A sensor function is used to determine the kinematic quantities required for a control force determination. The name of a sensor function begins with TS, usually followed by 15 characters. A force function is a function of time, generalized coordinates qi and generalized speeds ui.

A force function is used to calculate forces or torques that are exerted on the multibody system during simulation. The name of a force function begins with TF, usually followed by 15 characters. A user defined function is any of the functions, namely a time function, a Q function, a sensor function or a force function, which can be referred to as a user defined function unless it is explicitly defined in the preprocessor 13 . D or DD are added to the end of the chain of a user-defined function to determine the first or second derivative of the function over time.

The system receives input data in a symbolic or numeric format from a keyboard or port 14 or from previously created files. The input data are divided into four categories: (1) multi-body system definition data; (2) simulation data; (3) body shape data; and (4) terms for later availability. A multibody system consists of bodies, connections, limit conditions, drivers, forces, sensors and piece of information. All of this information is not applicable to every mechanical multibody system. Figures described hereinbefore. 1 is an example of a multi-body system which is exposed to an analysis by the analysis system and drive -ver that disclosed herein. As previously described, connections of a multibody system can first be disconnected by a modeler to define the bodies such that the system has a "tree" topology. The "loop" or motion limit conditions are introduced by the modeler at a later time. While it is in the unrelated "tree" topology, the system must be brought into a reference configuration (all qi = 0) and the fixed orthogonal unit vectors of three parts on each body must be defined so that in the reference configuration all baselines match . A body must be connected to another body by definition. The system offers a connection program library which contains the following connection types: in brackets; free (6 ° of a freedom connection); cylindrical; just; prismatic; in circulation; helical; spherical; universal and defined by the user. To define a body, a user must provide the following information: body name; Dimensions; Inertia; inner body (name of the previously defined body or multibody reference); from the inside for connection (position vector from the inner body mass center for connection in baselines of the inner body, three symbols); Connection to the body (position vector from a connection to the body mass center in the body baselines, three symbols). The limit condition can be entered by a modeler to define a "loop" topology and to impose motion limit conditions on the system. There are five types of limit conditions: position; Speed; Transmission; Cam; and defined by the user. To define a limit condition that is different from the user defined limit condition, a user must provide the following information: name; Type; Body-1 names; Body 2 names and parameters. As previously mentioned, the modeler must be knowledgeable enough to know how to define the holonomic generalized velocities for the disclosed analytical system. The modeler must supply the name of the limit condition, the type of the limit condition, and the functions of qt.

A driver is used to generalize the timeliness of a generalized Describe the speed associated with the connection. Thus a Modelers will be familiar with how generalized speeds for one Connection defined and arranged or entered. Becomes a modeler provide a driver with the following information: name; Connection name; the Number of degrees of freedom and the function. A program library from Kraftty pen is provided by the system. Currently in the program library included six types of forces. A taxpayer needs, for example assign a body 1 name, a body 2 name, a position vector from Body-1 center of mass to an intermediate point in Body-1 basic and then to the point of application or alignment in body-2 Baseline (six symbols), a position vector in the body-2 mass center to an intermediate point in the body 2 baselines and then to the point of application or alignment in the Body 1 baseline (six Symbols), a name of the force function, and a direction of the force on the body 2 in body 1 or body 2 baselines (three symbols). To define one Force, a user will specify the name of the force, the type of force, and the parameters such as those mentioned for control force herein is.  

The purpose of a sensor is to provide a measure of kinematic quantities that are required for steering force and torque estimation. These Sizes contain a position and a speed between two points, which are fixed on two different bodies, and one orientation and one Angular velocity between the two bodies. From a modeler to Guided parameters for the calculation of these quantities are required for each sensor type such. A translation sensor and a rotation sensor are provided. To the To specify a sensor, a modeler must name the sensor, the type of the sensor, a body 1 name, a body 2 name and the sensor parameter make available. As discussed hereinbefore, a modeler must Specifying a piecewise system a total number of "pieces" in the Provide the system. For each piece, the modeler is asked to provide a subset of temporal dependencies that are in the Multibody system are defined. The criterion for a transition between these pieces and how the transition is to be carried out by the sym bolic processor created to expressions for variables and parameters too generate that are required for the transitions.

Simulation data contain numerical solution parameters, user-defined subroutines and initial conditions and time parameters. Numerical solving methods are required for solving linear equations, ordinary differential equations and nonlinear algebraic equations from geometric limit value conditions. The parameters required for specification by a modeler for ordinary differential equations include a minimum integration step size, a tolerance, a fixed step size or a flexible step size. Nonlinear algebraic equation parameters contain the maximum number of iterations and the tolerance thereon. When defining the multibody system data, a modeler may specify the names of the functions instead of the explicit functions in order to describe limit value conditions, drivers, forces and sensors. These functions are then calculated during a simulation by calling a subroutine defined by a modeler. This subroutine is generated by the symbolic generator 15 and all functions are set to zero. A modeler must change this subroutine to implement the functions. Finally, a user or a modeler will determine limit conditions for q and u. The user will be provided with an equation solution to ensure that the specified initial q geometric or loop constraint conditions defined in the multibody system data. A simulation start time, an end time and an expression step size are also specified by a user. In order to view a multibody system before a simulation and to stimulate a simulated movement after the simulation, a user must define a geometrical shape for each body. The form is made available by the user of a file. The origin of each body is located in the center of mass, and the reference frame baselines follow that used in the reference configuration when the multibody system is defined. For the convenience of the user, a program library of simple geometric shapes is provided, which contains a box, a sphere, a cylinder, a cone, etc.

To postprocess the simulation data results, a user can perform thematic operations on the results before plotting. For example, the position, velocity, acceleration, vector, limit condition amount, etc. values can be obtained by commands to the post processor 21 .

The information generated by the disclosed system includes multi-body system definition data. A sufficient explanation for all data is created. This data can also be read by the preprocessor 13 upon request from a modeler. A further explanation is provided of how generalized coordinates q and generalized velocities u are defined. It should be noted that when all q = 0 are set, the multibody system has the reference configuration used to define the system. An input is also defined to the system data module 18 . This input contains a list of symbols to which values must be assigned and a list of function names defined by the user. In addition, an input to the symbolic generator 15 is defined. Certain subroutines are present in the system programs to find the position and orientation of each body for a given time and q, to calculate the sum of the squares of each remaining geometric boundary condition for a given time and q, the gradient of the sum of the squares of the remainder of each to calculate the geometric limit value condition taking into account q selected by a user for a given time and q and to calculate functions defined by a user which relate to the body position and the orientation and the geometric limit value conditions. The input data files 19 of FIG. 2 contain numerical values of all symbols, numerical solution parameters, initial conditions for a simulation and time parameters, a correspondence table between symbols defined by a program and symbols specified by a user, a topology of the multibody system before the limit value conditions and one of A key figure supplied to a user to identify the simulation cases. The simulation results from the code 20 executable by the simulation includes generalized coordinates q, generalized velocities u, derivatives over time of u, Lagrange multipliers, a mass center position, a velocity and an acceleration of each body, an orientation, an angular velocity, an angular acceleration each body, a sum of all forces acting on each body, including an addition from the Lagrangian multipliers and functions defined by a user. The data files 22 contain a time history of sizes specified by a user. A moving image excitation data file is included, along with a graphical representation of the multibody system, to allow a user to view the system prior to simulation and see how the system configuration changes when the values of q are changed. In addition, this helps the user select values of q as an initial estimate for solving the equations for a geometric limit condition. Graphic possibilities in the postprocessor 21 are used for the display. The data entry into the system by the modeler and the user are all checked by the system itself. Symbols entered in the preprocessor 13 are checked for a convention conflict. This includes words reserved for symbols, dual use of names for the multibody system elements, a function naming convention, and all multibody element conventions described herein. Numbers entered into preprocessor 13 and numerical values for symbols entered into the system data module by a modeler or user are checked according to the following rules:

• 1. The body mass must be greater than or equal to zero.  
• 2. The moments of inertia of the body must be greater than or equal to zero.
• 3. The sum of any two moments of inertia of a body must be be greater than or equal to the third.
• 4. Intersecting moments of inertia must meet physical possibilities suffices.
• 5. The size of a direction vector must be one.
• 6. All restrictions on parameters of a multibody system described herein stem elements would have to be observed.
• 7. Spring constants and damping constants must be greater than or equal to zero be.
• 8. Limit conditions and drivers must be independent of each other.
• 9. Initial conditions must be a geometric limit condition, one Limit condition when moving and a driver limit condition are enough.

Information inputs have been defined above and how information is implemented when it is processed by the system has been described. A description will now be given of the commands or capabilities provided to each system module for a modeler or user to define a multibody system and to monitor execution of the system software to meet the simulation purposes. As previously described, the system includes five functional modules: a pre-processor 13 ; a symbolic generator 15 ; a system data module 18 ; Simulation modules 19 and 20 ; and a post processor 21 . A typical procedure for an application by a modeler to implement the system functions includes the following:

• 1. Call the preprocessor 13 .
• 2. Define the body and connections.
• 3. Define the limit conditions.
• 4. Define the drivers.
• 5. Define the piecewise system.
• 6. Define the sensors and forces.
• 7. Retain the input and prompt the system to generate input files to the symbolic generator 15 .
• 8. Running the symbolic generator 15 in order to provide symbolic formats of mechanical system equations for a movement.

The modeler or a user then selects the process in the following sequence will follow:

• 1. Modifying subroutines defined by a modeler.
• 2. Compile simulation codes and position limit conditions in the text editor and compiler 17 .
• 3. Calling up the system data module 18 .
• 4. Define the mechanical system body or element shapes.
• 5. Enter the numerical values for the symbols and call up the data test ability to check the numerical values.
• 6. Watch the system.
• 7. Enter the initial values for q and check the configuration.  
• 8. Solve for initial values of q that with geometric limit conditions are compatible.
• 9. Enter initial values for independent u and check a singula rity.
• 10. Enter numerical solution parameters and simulation time parameters.
• 11. Storage or storage of the body shapes and requesting the system to generate input data files for dynamic simulation and the postprocessor 21 .
• 12. Run the simulation routines.
• 13. Calling the postprocessor 21 .
• 14. Enter a file name.
• 15. Entering expressions for mechanical system sizes of interest.
• 16. Plot the sizes of interest ( 23 ) and store the data in the file ( 22 ).
• 17. Add the mechanical system points of interest for image movement ( 24 ).
• 18. Select the mechanical system body for image movement and input the image motion parameter.
• 19. Moving the simulation results and / or generating an image movement data file.

A detailed description of the various system elements is now undertaken. Preprocessor 13 allows a modeler to define a multibody system and sequentially generate a Fortran (in this embodiment) code for checking data in system data module 18 and also an input file for symbolic generator 15 based on input from the modeler. The generated input file can also be read at a later time by the preprocessor for further modification. The pre-processor instruction structure is shown in Fig. 3 as an outline of the pre-processor capabilities. The FILE function relates to six sub-functions. The first sub-function NEW discards the model present in the preprocessor, if one exists, and opens a NEW file specified by the modeler to form a new model. The INCLUDE subfunction appends the model from a file specified by a modeler to the existing model, if there is one. The KEEP subfunction stores multibody data entered by the modeler. The STORE & GENERATE sub-function allows the modeler to manipulate multibody data defined by the modeler, which relates to generalized coordinates and speed symbols, which require assigned values and other inputs to the symbolic generator 15 . The DIRECTORY or SAVE sub-function shows all files currently in the preprocessor's memory.

The EDIT function allows selection of a mechanical system element before calling one of the five functions shown in FIG. 3. GENERATE allows a selected element to be created. CHANGE allows the parameter symbols and values of a previously defined element to be changed. DELETE allows a previously defined element to be deleted. COPY allows you to copy one element parameter to another. VIEW allows a visual view of all defined points of the selected element.

The HELP function provides help messages that are selected with a function or a subfunction.

ELEMENT provides a selection of all available aspects for defining a mechanical multibody system. Seven such selections can be seen which are assigned to the ELEMENT function in FIG. 3.

The COMPOSITION function allows you to check the mechanical Many-body system definition and generates the generalized coordinates q and the generalized speeds u. A subfunction TREE ABOUT CHECK allows a construction of a "tree" topology arrangement of the defined one Model too. A subfunction Q & U DEFINE an assignment of q and u to each connection after changes in the body and the connection. LIST Q & U displays the definition of q and u too if no change after the last Q&U DEFINE by has been conducted. DOUBLE SYMBOLS provides a list of all symbols Which have been used more than once in the system definition and lists the mechanical system elements where they were used. SYMBOLS SHOW provides a list of all symbols used in mechanical System definition have been used. SUMMARY presents an advertisement the total number of points available for each multibody system element have been defined. If the definition of q and u is straight through is then a display with the total number of q and the Degrees of freedom of the multibody system.

The DISPLAY function provides a choice for the background display of the system. The BODY TREE subfunction provides the "tree" topology of the system, including the connection names. An example of the "tree" topology is shown in FIG. 4. If a connection name appears in a body definition but has not been defined as a connection element, there will be an asterisk in front of the connection name. The symbol N always stands for an initial reference frame. The chain feature of the system is displayed in the horizontal direction and the branch feature is displayed in the vertical direction of FIG. 4.

The purpose of the symbolic generator 15 is to read the data file generated by the preprocessor 13 and to generate a symbol / function list file; two subroutines were used by the system data module 18 and a simulation program in the FORTRAN language.

The purpose of the system data module 18 is to provide modelers and users with means for entering numerical values of symbols, initial conditions and time parameters for a simulation. Physically impossible numerical values are checked as previously described herein. The initial conditions are also checked for whether geometry and motion limit conditions are met, and upon request of the modelers and users, solutions to the limit condition equations for the initial conditions of the generalized coordinates and speeds selected by the modeler or user are available posed. A kinematic singularity will also be detected and the system will automatically select independent generalized speeds to solve the problem. The system allows a modeler or user to shape bodies and create bodies graphically selected from a display. The position and orientation of each of these selected bodies can be changed according to generalized coordinates entered by a modeler or user. The values of the generalized coordinates can be changed using the keyboard 14 of FIG. 2 or other device, if one is available. In this exemplary embodiment, all calculations relating to the body position, the body orientation and the limit value condition equations are carried out using FORTRAN codes which are generated by the symbolic generator 15 . Therefore, the FORT RAN codes have to be changed when a time which prescribes kinematic functions has to be renewed, and then they have to be compiled. The compiled target code can then be connected to the system data module 8 . Alternatively, a modeler or a user can connect these FORTRAN object codes in order to form a stand-alone executable code which can later be introduced or set up or initiated by the system data module 8 . This requires that the symbolic generator 15 generates a FORTRAN main program instead of mere subroutines.

The system data module 18 functions in the following manner. A FILE function contains five subfunctions, as can be seen in FIG. 5. A sub-function OPEN opens a file generated by the symbolic generator 15 , which stores all symbols and user functions that are used to define the mechanical multibody system. In addition, the previously created simulation data file can be opened if it is specified, and the previously generated body shape file can be opened if it is specified.

A KEEP FORM OPEN subfunction keeps the one just defined Body shape in a file. KEEP SIMULATION DATA ON the simulation data just defined, the symbol values, parameters and Initial conditions included. This file is used as input for the simulation program used. A subfunction DIRECTORY shows all files with one Data entry / exit file extension in the directory. A subfunction EXIT allows a medellator or user to leave the system Data module with a confirmation regarding data that is not saved are to get out.

The SYMBOL function allows a modeler or user to numeric Enter or change values of all listed symbols, namely on a type that is similar to screen editing. An explanation of each symbol can be provided on the screen at the request of the user will.

CHECK SYMBOL allows all entered numerical values to be checked by calling the data check subroutine generated by preprocessor 13 as mentioned hereinabove.

OBJECT allows creation of objects to the shape of a mechanical Define system body. The body shape is used for display purposes and can be used for later image movement of simulation results.

PARAMETER allows a modeler or user to enter parameters or change the accuracy of numerical solutions and a simula tion time.

START allows you to enter or change the initial conditions of q and u in a way that is similar to screen editing.

A GEOMETRY subfunction provides a calculation of the geometric limit condition equation deviations and a change in values from one Users selected dependent q available that the geometric Limit conditions are sufficient if the user requests it.  

A sub-function MOVEMENT allows the movement limit to be calculated deviations in value conditions and a change in the values of a user selected dependent u that impose the motion limit condition are sufficient, at the request of a user.

A DISPLAY function contains five subfunctions, as can be seen in FIG. 5.

GROUP allows a selection of bodies from the mechanical system is shown.

LOOKING creates a different view of the group.

DRAW allows the selected body to appear on the screen according to the Initial conditions of the q is drawn.

ZOOMING allows the user or modeler to section a section enlarging scene defined.

MOVE allows a modeler or user to generalize Select coordinate and the mechanical system according to a given Increment of the value of the selected q to move, on request of the user, using a mouse or the keyboard. In addition, this allows Subfunction a user, the initial conditions in the just existing Change values.

The simulation code generated by the symbolic generator 15 and the simulation data generated by the system data module 18 are both required for a simulation. A modeler or user must follow the following procedure for a simulation.

• 1. Changing the subroutine, which is generated by the symbolic generator 15 , with respect to the forces and / or torques.
• 2. Compile and link the simulation code to generate an execution codes.
• 3. Run the executable code.

The purpose of the post processor 21 is to allow a modeler or user to manipulate simulation result data, create data files containing selected quantities in a format specified by a user, plot xy data, and move the image of the simulation results. The function of the post processor is shown in Fig. 6 of the drawing pages. The FILE function refers to five sub-functions, as can be seen in the drawing. An OPEN sub-function provides the opening of a simulation result file. USE activates a previously opened file. CLOSE closes a previously opened file. DIRECTORY shows all files with extensions related to the postprocessor. EXIT allows the process to exit the postprocessor. A HELP function displays help messages that belong to selected sections of the procedure.

The INFORMATION function affects seven sub-functions. FILE allows the names of the files that have been opened to be displayed and highlights the file that is currently in use. HEAD shows the head that is defined in the active simulation result file. TOPOLOGY displays the mechanical multibody system topology in the same manner as previously described for preprocessor 13 herein. A sub-function INPUT shows the multibody system definition. Q & U show the definition of the generalized coordinates and the generalized speeds. A subfunction USER FUNCTIONS lists all names of the functions defined by a user. SIMULATION DATA shows the numerical values of the symbols and the simulation parameters.

The DATA function affects four sub-functions. A sub-function OFF PRINT allows modelers and users to create their own variables Define vector or scale. An EXTERN sub-function allows the modeler or user, the variables by reading values from one define external file. A SHOW sub-function displays  numerical values of a variable in one by a modeler or user the specified area or framework is available. DIRECTORY allows one Display of all variables defined by a modeler or a user and their definitions.

A PLOT function affects seven sub-functions. A subfunction Curvature allows the user to define curvatures caused by a Screen editing can be plotted. A WINDOW sub-function allows this Processor to select a window to cover all the data and so far spread as possible. The user can also have a window and a number of divisions in x and / or y. A GRID subfunction allows a modeler or a user to switch grid lines on or off. TITLE allows a modeler or a user to determine a title men. A LABEL X and a LABEL Y allow a user to have a label determine. A sub-function DRAW represents the drawing of the defined one Curvatures are available on the display screen. A function SCREAM BEN contains two sub-functions. ASCII allows modelers or users temporal histories of a selected variable in a particular ASCII Writing format. RESTART allows the creation of an input file for a simulation for a specific model or user graced time to restart.

The ANIMATION or MOVE IMAGES function shown in Fig. 6 relates to three functions. A BODY / POINTER SELECTION sub-function offers a modeler or user a selection of bodies or pointers to be included in the image movement. A SELECT subframe function allows a modeler or user to select a range of frames and allows jumping between those frames during image movement. MOVE IMAGE starts an image movement on a display screen.

A confirmation made by the system disclosed herein answers the question as to whether calculations of the system have been successfully implemented. Certain performance limits are inherent in the system as follows. The complexity and size of the mechanical multibody system that can be treated depends on the available memory. In the preferred embodiment described herein and in the best known manner, the system is limited to 50 mechanical system bodies or elements, 100 generalized coordinates (degrees of freedom), 100 limit conditions and drivers, 100 user-defined functions and 600 symbols . These numbers serve as a target complexity and do not necessarily reflect the limit conditions due to the memory size of a particular computer system. Data entered by modelers and users through the preprocessor 13 , the system data module 18 and the postprocessor 21 are checked against convention conflicts, physically impossible arrangements or properties and oversize of the defined mechanical multibody system.

The software that is designed to run the analysis system described herein is to any computer system with a UNIX operating system, a standard C compiler and a driver / library for PHIGS (or X-WINDOW) transportable. The simulation program generated is as herein previously mentioned, on-site or remote therefrom to any disclosed herein Transportable analysis system using a standard FORTRAN-77 compiler Has.

Referring to an example of the operation of the disclosed analysis system, reference is made to FIG. 7 of the drawing. Fig. 7 shows a Stanford arm which consists of six elements or bodies, which are denoted by B 1 , B 2 , B 3 , B 4 , B 5 and B 6 . B 1 can be rotated in a Newtonian reference frame N about a vertical axis n₂, which is fixed at both N and B 1 and B 1 supports B 2 , which can be rotated relative to B 1 about a horizontal axis, both is fixed at B 1 as well as at B 2 . B 3 is supported by B 2 and can be made to translate relative to B 2 while carrying B 4 , which can be rotated relative to B 3 about an axis fixed to both B 3 and B 4 . B 5 is connected to B 4 , and B 6 is connected to B 5 in such a way that each can be rotated about an axis relative to the body carrying it, which axis is fixed in both the supporting and the supported bodies .

To run the Stanford arm FORTRAN program, a must Modelers first generate executable code 03218 00070 552 001000280000000200012000285910310700040 0002004118302 00004 03099 and then a data file generate that contains the following information or commands:

• 1. A description of the run (in one line).
• 2. The numerical values of the following parameters (system dimensions of interest):
  L1, L2, L3, L4, L5, L6, MA,
  IA1, IA2, IB1, IB2, IB3, MC,
  IC1, IC2, MD, ID1, ID2, ME, IE1,
  IE2, MF, IF1, IF2, IF3, G, K
• 3. The initial values of Q1, Q2, Q3, Q4, Q5, Q6
• 4. The initial values of U1, U2, U3, U4, U5, U6
• 5. The choice of the integration process.
  1: Fixed time step.
  2: Flexible time step.
• 6. The simulation start time and end time.
• 7. The integration time step.
• 8. k: Results will be printed every k integration time steps.

Example 1 (in the system data module)

1 2 3 4 5 6 7
8 9 10 11 12 13 14
15 16 17 18 19 20 21
22 23 24 25 26 27 28
0 0 0 0 0 0
1 1 1 1 1 1
2nd
0 5 0. 1 1

System parameters (simulation codes / documentation echo of the input):

L1 0.100000D + 00
L2 0.100000D + 00
L3 0.700000D + 00
L4 -0.100000D + 00
L5 0.000000D-01
L6 0.140000D + 00
MA 0.900000D + 01
IA1 0.100000D-01
IA2 0.200000D-01
MB 0.600000D + 01
IB1 0.600000D-01
IB2 0.100000D-01
IB3 0.500000D-01
MC 0.500000D + 01
IC1 0.400000D + 00
IC2 0.100000D-01
MD 0.100000D + 01
ID1 0.500000D-03
ID2 0.100000D-02
ME 0.600000D + 00
IE1 0.500000D-03
IE2 0.200000D-03
MF 0.500000D + 00
IF1 0.100000D-02
IF2 0.200000D-02
IF3 0.300000D-02
G 0.981000D + 01
K 0.000000D + 00

Initial condition:

Q1 = 0.
Q2 = 0.
Q3 = 0.
Q4 = 0.
Q5 = 0.
Q6 = 0.

U1 = 0.
U2 = 0.
U3 = 0.
U4 = 0.
U5 = 0.
U6 = 0.

Integration with flexible time step (echo of commands):

Start time = 0.
End time = 5.00000
Integration time step = 0.100000 seconds
Result that is printed out every k steps; k = 1 (here: every 0.1 seconds)
Issue selection:
System description
Mass center position - global
Any body orientation - global
Mass center speed - global
Any body angular velocity - local
Acceleration - global
Angular acceleration - local
Simulation check
Active force - global and torque - local
Chronological history of Q
Chronological history of U
Chronological history of UD
Movement image file, topology information
Original numeric user input file

Although here the best form for carrying out the present invention viewed, shown and described, it is obvious that a variation and variation can be carried out without deviating from what as Subject of the invention is considered.

Claims (31)

1. A method for effectively analyzing a holonomic, non-holonomic or partially non-holonomic multibody system, characterized in that
the topology of a general type of multibody system is defined,
symbolic equations are generated, which represent the equations of motion of the general multibody system,
numerical values are entered which define a specific multibody system within the general type,
the operation of the multibody system is simulated,
Simulation results are obtained and that
the acceptability of physical and system-imposed limit value conditions with regard to the entered numerical values is checked and displayed. 2. The method according to claim 1, characterized in that that defining the topology includes the step of Define basic open loop topology. 3. The method according to claim 1, characterized in that defining the topology includes the step of one Define tree topology. 4. The method according to claim 1, characterized in that defining the topology includes the step, limit to define which closed circular Topology and non-holonomic constraints defined.   5. The method according to claim 4, characterized in that defining limit conditions the level includes defining combinations of limit equations which are used at different times, to de a partially non-holonomic dynamic system finish. 6. The method according to claim 1, characterized in that determines the initial conditions of the appropriate system will. 7. The method according to claim 1, characterized in that the simulation results are manipulated to ver to get different possible system sizes. 8. The method according to claim 1, characterized in that the time course of the simulation results ge is saved. 9. The method according to claim 7, characterized in that the time course of the sy stem sizes is saved. 10. The method according to claim 1, characterized in that the resulting simulation results in two Dimensional format are applied, so that an overview a large amount of information is created. 11. The method according to claim 1, characterized in that the movements of the multibody system are stimulated, so that vivid visual information is obtained becomes. 12. The method according to claim 1, characterized in that that the simulation of the simulation system includes the stage  to solve ordinary differential equations and the solutions to examine integral results to determine the accuracy of the to determine symbolic equation formulation. 13. A method of analyzing a mechanical system comprising interconnected mechanical elements, wherein
the basic topology of the mechanical system is defined,
an inertial system with a fixed reference is selected for the mechanical system,
the limit conditions of the mechanical system are specified, and
symbolic equations of motion for the non-free system for a holonomic, non-holonomic and sometimes non-holonomic dynamic system are generated from the specified system constraints. 14. The method according to claim 13, characterized in that the time course of a generalized Ge speed in association with a connection between Elements in the mechanical system. 15. The method according to claim 13, characterized in that the types of force are defined which are based on the mecha act on African elements. 16. The method according to claim 13, characterized in that scalar kinematic quantities for system force and rotation torque calculation can be specified. 17. The method according to claim 13, characterized in that  the availability and accuracy of the generated symbolic equations regarding system convention and restrictions are checked and displayed. 18. The method according to claim 13, characterized in that that created a standalone executable code is, which the changeability or portability of ver delayed execution on site and off site. 19. The method according to claim 13, characterized in that
the numerical values for symbols and initial conditions are entered in order to obtain system-specific equations of motion, and
the system-specific equations of motion for body position, orientation and limit value condition equation simulation results are solved. 20. The method according to claim 19, characterized in that the simulation results are manipulated to pre to get certain system sizes into consideration. 21. The method according to claim 19, characterized in that the time course of the simulation results ge is saved. 22. The method according to claim 20, characterized in that that the timing of the system under consideration sizes is saved. 23. The method according to claim 13, characterized in that that the simulation results in two-dimensional format be applied so that an overview of a considerable Chen amount of information is obtained.   24. The method according to claim 13, characterized in that that who animates the movements of the mechanical system the so that visual information is obtained. 25. Device for analyzing a mechanical system, which has interconnected mechanical elements, characterized by
a preprocessor,
a data input device for inputting the mechanical system topology and limit value conditions into the preprocessor,
by means of a device coupled to the preprocessor for formulating symbolic equations of motion for a holonomic, non-holonomic and sometimes non-holonomic, non-free mechanical system based on the entered topology and limit value conditions. 26. The apparatus according to claim 25, characterized in that
means for entering numerical symbol values and initial conditions into the equations of motion is provided, and
a device creates solutions for the equations of motion with the entered numerical values and initial conditions in order to obtain simulation results which indicate the position of the mechanical element, orientation and limit determination equation simulations. 27. The apparatus according to claim 26, characterized in that a device for manipulating the simulation results nisse is provided so that predetermined come into consideration system sizes can be obtained.   28. The apparatus according to claim 26, characterized by a device for recording the simulation results nisse in a two-dimensional format. 29. The device according to claim 26, characterized by a device for animating the movements of the mecha African elements. 30. The device according to claim 25, characterized in that that a device for validating topology and Limit conditions of the mechanical system regarding System convention and physical possibilities featured see is. 31. The device according to claim 26, characterized in that that a device for validating symbolic numeri values and initial conditions regarding System convention and physical possibilities featured see is.