JPH0276690A - Robot arm track control method - Google Patents

Abstract

PURPOSE:To compensate a non-linear term accurately by making coarse control using the nominal value of physical parameter of an arm, representing then a virtual value by this nominal value and the upper limit value for varying portion therefrom, and giving the portion, which the previous control can not follow up after target track, the mentioned upper limit value for varying portion from the nominal value of physical parameter. CONSTITUTION:The condition of system is brought to a certain changeover surface, and after attainment of the changeover surface, control is made by the use of sliding mode to make convergence to the point of origin while sliding on this changeover surface. Therein the behavior of the system is restrained by the changeover surface and decided unambiguously only upon the characteristic of the changeover surface. Because this characteristic is not dependent upon turbulence or physical parameters of the plant concerned, there is no need to previously know the physical parameters of the robot arm and the load applied thereto in order to apply controls.

Description

DETAILED DESCRIPTION OF THE INVENTION [Field of Industrial Application] The present invention relates to a control method for causing a robot arm to respond to a specified target trajectory at high speed and follow it without error.

[Prior art] Conventional robot arm control + Ii b needle methods are often based on the idea of linearization of mathematical models, and nonlinear forces such as gravity, centrifugal force, and Coriolis are compensated for by feedforward control. was. Since these forces depend on the physical parameters of the robot arm and the load applied to it, feedforward compensation could not be applied unless the physical parameters and load were accurately known in advance. In addition, if the load changes, the control law must also be changed in order to maintain the original control performance, so this is not a practical control when assuming work contents such as industrial robots where the load fluctuates. Ta.

In addition, in a normal robot arm, it is inevitable that model errors will occur in the resulting mathematical model due to simplification in control design, uncertainty in physical property values, and the influence of nonlinear characteristics. There were problems with accurate compensation.

[Problem 1 to be Solved by the Invention In order to solve the above problems, the present invention uses sliding mode control. Sliding mode control is characterized by bringing the state of the system to a certain switching plane, and after reaching the bending plane, using discontinuous inputs, the switching plane springs up and converges to the origin.

In this case, the behavior of the system is constrained by the switching surface and is uniquely determined only by the characteristics of the switching surface. Since the characteristics of this switching surface are independent of the physical parameters of the plant and disturbances, the control performance is robust to them. In other words, there is no need to know in advance the physical parameters of the robot arm or the load to be applied to it in order to perform control. Furthermore, control performance is also independent of variations in the physical parameters of the robot arm and the presence of disturbances.

[Means for solving problems]

As is well known, the equation of motion of the robot arm is that the angle vector 9 between each link is defined as θ=[θ1. θ2.・
..., θ,,]■, then M (θ) i! l+B (θJ)+D /j+G
It is expressed as (θ)=u+portrait (1). Here, the meaning of each symbol is as follows.

M(θ): An inertia matrix determined by the attitude θ, and is a fixed symmetric matrix.

B (a, tj): Centrifugal force and Coriolis term, two viscous friction force terms D = d l a g (D + , D 2+")
+ D n ): Viscous friction coefficient G (θ): Gravity term U: Input torque W between each link: For example, represents disturbance such as static friction. What should be noted is that the customer element of the inertia matrix of a general robot arm are the physical parameters link length, center of gravity position, mass, moment of inertia regarding the center of gravity position, and link angle sin, cos(
Since it is a combination of addition, subtraction, and multiplication of Ii, it has upper and lower boundedness. The other terms, centrifugal force, Coriolis, and gravitational coefficients, are also upper and lower bounded.

Hereinafter, the control law of the present invention will be derived based on the equation of motion of equation (1).

The target trajectory and target tracking speed are /j a and II, respectively.
It is written as a. Then, the angular error e(t) and the angular velocity error (t) are respectively e(t) = θ(1) - θa(...(2) center (
t): Yu (1) −〇, + (1
) ... (3). Also, the switching of n +II and s, = O(i = 1, . . .
, n) are collectively written as a vector in the following equation.

S=[S++ S2+...+s,]T However, Sl is S, =c, e, +4. ．． Define a,>0. Therefore, s = Ce + b = (4a
) C= d i ag (CI, C2, ·, C,) =
...(4b).

In order to converge the state of the system from an arbitrary initial state to the switching surface, we use the Lyapunov function V (t) = (1/
2) Choose s'M (#)s - (5). At this time, V
If (t) < O (s≠0) is satisfied, t→■
Then, V(t)→0, and the convergence of the system state is guaranteed. However, the upper-key symbol °゛・′ means time differentiation. Therefore, V(t) is V(t) = (1/2)s'M(61)s +
s''M(a)A=(1/2)S'M(IJ)s+s
TM(θ)(Ct4+ri=(1/2)s'
M(θ)s+s'[M(θ)C6+M(θ)l
j−M (/7) #v]... (6) As can be seen from equation (1), M(θ)/J=u+w−B(θ, θ) −Dυ−G(θ) ・...Write (7) and substitute this into equation (6).
(θ)C! +u+w−M(θ>IJ ,,+
B< θ, C)-D θ-G (θ)]... (
8) Become. In equation (8), if M(θ
) C6+ LJ+WM (θ)b.

-B (G, d)-Dθ-G(θ)=-P(t)s...
・(9) If P (t)=diag [P:(t)], what is 9(t)? (t)=(1/2)s"ro'!(θ)s -s'P
(t)s... (10) Then, p:(t) becomes p:(t)>ΣJI Mt)/2 1 +,
−(11)k,>O: When taken as an arbitrary positive constant, the following equation is returned from equation (10).

V (t) = -s'd i ag (LC+) S
...(12) According to Geshgorin's theorem, the second term on the right side of equation (12) is negative definite. Therefore, by setting V(t)<-sTdiag(k, n5---(L3)), v(t) becomes negative definite, and it is guaranteed that the state of the system converges on the switching surface. S on the face
When =O is reached, as shown in equation (4), a, >
0. Since i=1 to n is set to be stable, convergence to the origin can be guaranteed, and thus the stability of the entire system can also be guaranteed. In the following, a control law U that satisfies equations (9) and (11) is determined.

Now, the true values of the physical parameters M , B , D , and G are expressed by their respective nominal values and variations therefrom, as shown in equation (14).

M(θ)=M"(θ)+7M(θ) B(C1)=B"(θ, θ)+ΔB(θJ)D=D"
+ΔD...(14)G (θ)
=G''(θ)' + A G (θ) However, A6 represents the nominal value of A, and AA is the variation from the nominal value.

Next, the upper limit value is represented by an alphabetic character with "°". That is, lJM151 ≦θs1 Δ B, l ≦0゜l A D 11 ≦0.
... (15) l A G, l ≦C1. The input is divided into two parts and set as Uwa+ΔU (16).

However, 0 is C1=B” (θJ)+D” θ0G” (θ)−
Define M(θ〉0 case...(17). Therefore, finding (1) is equivalent to finding I!lu. In equation (9), (14), (16)
, (17) and take out the first element of U to obtain equation (18).

Au, +w knee M: (θ)a, -dB: (θ, etc.) -I
ID, θ, -74G, (θ) + #M, C, 4 = -p
:(t)S, ...(18)S,≠
0, if both sides of equation (18) are removed by -8, -[du, +w, -M, (θ), a, -7B, (θ
, υ) −iJ D 1, −, 6G, (θ) + A
M, C combination]/S1= p 1(t)
...(19) is obtained. From the condition of equation (11), −[d u , +w, −M, (θ) #d−ABI(θ,
υ) −aD, li nee aaAe> +aM, cal/s
.

≧Σ1 M, /2 1 + k, ・・
・(20) holds true. To rearrange this, [tu, 1S1 in ten]/S1 ≦ [-w, +M1(θ)a, +aB, (θ, θ)+-I
D, θ, −111M1c! ]-1-zlG+(θ)
]/S, Σ・l M Ip/ 2 1
...(21). Here, the disturbance W1 and the target angular acceleration θ can be assumed to be bounded without loss of generality. Since θ4 is bounded inertia matrix M(θ) is also bounded above, M.

(θ)θ is also bounded. Therefore, Wl and M, (θ
)i, are written as ・sin and Ql, respectively. That is, IW口≦Kai1M1(θ) lJ, I≦9. ...(2
2) I M Schif1 ≦F>1+). Here, M, (8) is the first row vector of M(θ). Since the inequality in equation (21) always holds true, the left side of equation (21) takes a negative value of the upper limit of the right side of the open equation! Determine luI as shown below.

[/3u: +1 sl/s, = -[V/, + 9: + B: (θ, θ) 10 G1 (
θ)+ΣJ(MC)+)I White 11+ Dll 0.1
] / l s , 1・-ΣJM 14/2
...(23) That is, using the relational expression l s l sgn (s) = s, 7! lu, A u , = −k , s , −Σ4 M 11 /
2 S knee S g n (S 1) [@, + U,
+@, < θ, b > + t-l < θ) + Σ
, (Mc), Jl combination, l+D, l θ11 ]・
...(24) If we take this, the inequality in equation (11) always holds true.

However, (rG'1c)11 represents the element in the i-th row and j-th column of IQc. Therefore, the stability of the system is guaranteed according to Lyabunov's stability theory. At this time, equation (24), (1
According to equations 6) and (17), the control law u1 of the present invention is finally u, = B'', (θ, θ) + D', θ + G'', C&)
-Mlo (θ) C@k: S I-2w I
VI; J/2s knee sgn (Sl) [
10, +9: < θ, θ) 10 G, (θ) + Σ・(Simplified C
)ld combination 11 + death・11 θ, 1]... (2
5) It becomes. That is, the control law of equation (25) is based on the nominal values B°, (θ, θ) of centrifugal force and Coriolis force, and the nominal value D0. θ, the nominal value of gravity jmG”, <θ), 1M which is the nominal value of the inertia matrix multiplied by the coefficient of the switching surface and the angular velocity error (θ)CΦ, the product of any positive constant and the state of the switching surface. s,, the product of half the sum of each upper bound value of the inertia term 1-row element and the state of the switching surface - Σ4r'A L)/2 s.

Each addition value, the upper limit of disturbance 1, the upper limit of the product of inertia matrix and target angular acceleration 91, the upper limit color of centrifugal force and Coriolis force, (θ, J), the upper limit of gravity C, (
θ), the upper bound of the inertia matrix multiplied by the coefficient of the switching surface and the absolute value of the angular velocity error, Σ, Ml; IIC), + e:l, and the upper bound of the viscous friction coefficient (a[), of the angular velocity The upper bound of the viscous friction force multiplied by the absolute value Iθ11 f directfi, l J,
1. Each addition is made up of the sum of amounts that are switched between positive and negative depending on the sign of the switching surface. The configuration when this control law is applied to a robot arm is shown in Figure 1. In the figure, the state quantities of the robot arm that can be measured are the joint angle θ and the joint angular velocity 49.

From these measurements, the errors expressed by equations (2) and (3) are calculated, respectively. However, as expressed in equation (25), compensation is applied by performing calculations using the nominal values of physical parameters in real time.At the same time, the current state of the robot arm is defined by equation (4). It is monitored by a switching variable, and the overall control is completed by switching the addition amount of each upper limit value to positive or negative depending on the sign of this switching variable.

Now, the control ff1j in equation (25) shows that control can be achieved without using accurate physical parameters, but if applied as is, if the omitted high frequency Dynamics, if present, can be stimulated and cause undesirable responses such as mechanical resonance. Therefore, an effective control method for the case where high-frequency dynamics is present and is not included in the mathematical model of the robot arm.
is found below.

As can be seen from equation (15), the input is discontinuous, but
The reason for this is that the sign function sgn(s) is used. When the state of the system switches and enters near s, = O, the input becomes discontinuous due to sgn(s+),
It fluctuates wildly. When it is actually applied to a multi-link arm system, it is thought that it will excite high-frequency dynamics parts that are not modeled in the system, such as mechanical resonance, which may cause oscillation. Therefore, in order to make the input continuous and avoid undesirable responses such as mechanical resonance, the continuous saturation interval number sat (
s/ε) can also be used. That is, using 5at(s/ε) instead of sgn(s) in equation (25),
Human power u5 is u, = B, ″ (θ, θ) + D θ, 10G1°
(θ)−M,. (θ) Ch k : s 1
-ΣJ+VI L4/ 2 S l-5at(S, /6
1) [gV, +Ql+Ell(θ, θ) 10, (θ)+
ΣPacking C) L, (1+4Il + D+I force Lo]...
(26). The control system configuration when executing this control law is the same as that shown in FIG. In the case of equation (25), the function that switches the amount of addition of each upper bound value was a sign function, but (26)
In the formula, it is simply a saturation function. Now, s
The characteristics of the gn(s:) and 5at(S,/εl) functions are shown in FIGS. 2 and 3. As can be seen from both figures, i
When s, l>ε, sat (sI/ε:)=sgn
(s:) Therefore, the input of equation (26) is (25)
Matches that of Eq. Therefore, convergence in the direction of the cutting plane when ls:l>61 is guaranteed. However, Is,
When l<61, 5at(s:/ε1)≠sgn(s:
), convergence is not guaranteed, and there is a possibility of repulsion on the bending surface. a1 Overall, only the convergence of the system state to a thin layer 1s11≦61 as shown in FIG. 4 is guaranteed. In this case, the input is continuous, but the drawback is that errors also occur. This error is estimated as follows.

First, it is assumed that the state of the system is in the thin layer 1s, I≦81. That is, if si=ξ(t), (ls(t)1≦εI), then (
4) According to the formula, S 1” CHe ++ 61=ξ(t) ・ (
27) Therefore, when 1=0, the initial value is e(
0) = 0, the solution to equation (27) is 6, =, r
'Hξ(r) e x p (-c 1(t-r
)) d r. Therefore, Ie, li/: ξ (r) e x p (-c + (t
−r))d τ1≦ /:1ξ(r)I exp(7c
:(t-r)) aτ≦ dog ε1e x p (-c:
(-τ))dτ≦ε+/CI(1-exp(-c:t
))≦ε1/C1...(28) It can be expanded as follows. That is, the error can be suppressed to within ε,/c1. Therefore, if cl is set large, the convergence to the target value becomes faster and the error can be reduced. If B1 is made large, the human power becomes smoother, but the error becomes larger. Therefore, when actually using the control law of equation (26), it is necessary to consider the trade-off between convergence speed and error.

[Operation] Since the control fJ1 law of the present invention is expressed by mathematical formulas as shown in equations (25) and 26), it is difficult to explain its operation strictly, but the control law of the present invention can be easily explained. The effect of this is explained as follows. That is, rough control is first applied using rough values, ie, nominal values, of the physical parameters of the robot arm. Since this approximate value is different from the true value, it is difficult to follow the target trajectory with the current tell fH. Therefore, when the true value of a physical parameter is expressed as the nominal value and the upper bound value of the variation from the nominal value, the part that could not follow the target trajectory due to the previous control is expressed as the upper bound of the variation from the nominal value of the physical parameter. A value is given and used for control and correction.

[Example] The results of applying the control law of the present invention to a two-link arm as shown in FIG. 5 will be shown. The equation of motion of this two-link arm is M (θ)#+B Ce as shown in equation (1).
, lI) + DJ + G (/?) = U 10, and the physical parameters M, G, and B are as follows.

M11=J)+J2+m1. r2
2+ m2 (r2' + 112)+2 m21
, r2c o sθ2 M, 2=M2. = m2r2”+ J2+ m21,
r2c o sθ2M 22 = m 2 r 22+
J 2B+(61,θ)=-2m21. r2 sin θ
2# +72-m21, r2s inθ2θ22B
2(θ, θ) = m21, r2S i nθ2sha, 2
G, (θ) = (m, r, +m21t)gcO5θ.

+m2r2gcos (θ1+θ2) G, (θ) = m2r, gcos (θ1102) However, ml, m2: jtfltJ of links 1 and 2,,
J2: Moment of inertia 1, . about the center of gravity position of link 1.2. 12: Link length r In of links 1 and 2
r2: Center of gravity position g of link 1.2; gravitational acceleration. Here, the nominal faB''(θ, θ) of equation (25)
Let's set t M@(θ)Ce as B' (θ, θ)=O M'' (θ)C sum=0.In other words, the nominal value is set to zero, and all the errors that occur are included in the fluctuation. In this case, El, (θ, θ), IAic becomes the upper bound of that term, 9, (θ, J) = (m,, l + r2), , K(2I
Shio, θ, I + θ22) B2 (θ, a > = < m 211 r 2) 1
111-012M+CI A'l=M++CII"
+I +M+2C21 white, l= [J, +J2+m, r
2"+m2(r2'+ 1 +")+217121+2
”21 ++a+ell ee1+ (m2r22+
J2+m21, r2) 11. , c 21 e2t
M2CI White l=M, 2c, l White, 1 + "
Cut 22C2j, 1= (m2r 22+ J2
+m21 +r 2) na+c 1 l e+I
ten (m,, r 22+ J 2 n ma
: r c 2 + 6 2 1 can be taken.

D", G" (θ) take the identified values of physical parameters. At this time, 6(θ), 0 means an identification error.

In order to verify the control law of the present invention, numerical experiments were conducted using the two-link arm shown in FIG. 5, which has three links 1 and four links 2. The nominal and identified values of link length, center of gravity, mass, moment of inertia, and coefficient of viscous friction used in the experiment are shown in the table below. As shown in the table below, approximately 10
We are taking percentage differences.

Table 2 True value and nominal value of physical parameters of link arm Also, as a disturbance, a random number function with a face maximum value of 1 is applied to the robot arm side. Below we show the results of numerical experiments.

Figures 6 and 7 are linear trajectories using the control law of equation (25): x = 0.3 [m] y = 0.2-
This is the input and tracking error when tracking 0.2t [ml (t<2sec). As shown in FIG. 7, the error almost converges to zero, but the input in FIG. 6 fluctuates wildly.

FIGS. 8, 9, and 10 show the pattern of linear tracking when the control law equation (26) is used, and are stick diagrams, input, and tracking errors showing the state of linear trajectory tracking, respectively. Here, c,=c,,=5. ε, = ε2±0.02. Since the −1 input is continuous, smooth operation can be achieved without exciting mechanical resonance or the like. However, a tracking error occurs.

Figures 11, 12, and 13 are circular orbits: x! 0.27+0.115sin (3,14t)
[mlylI0.115+0.115cos(3,14
t) [ml (t<2sec) i.e. (x-0,27)2+(y-0,115)21IO,1
152, and are a stick diagram, human power, and tracking error showing the state of circular orbit tracking, respectively. However, c ,= c 2= 3 , ε1=ε2=
It is set to 0°05. Although there is some tracking error, it traces a smooth circular orbit.

The results of numerical experiments demonstrate the effectiveness of the present control method and its robustness against parameter variations and disturbances. Also, the 9th
As is clear from the figure and FIG. 12, when the saturation function was used, the input was almost continuous. Therefore, it is possible to stimulate the high-frequency dynamics part of the robot arm whose modeling is omitted, and to make the robot arm follow the target trajectory without exciting mechanical resonance.

In this embodiment, the control performance was shown using a two-link arm as the control target, but the same control performance can be obtained even in the case of a multi-link arm.

Furthermore, the control law of the present invention is applicable not only to robot arms but also to control of general mechanical systems such as positioning tables, electric motors, etc.

[Effect of the invention] The effects of the present invention are summarized as follows.

(1) Robust to physical parameter fluctuations and disturbances of the robot arm.

(2) Since the control law does not require inverse matrix calculation of the inertia matrix, the design is very simple and the control law is easy to implement. In other control design methods, the inverse matrix of the inertia matrix M-1
The control law becomes complicated because it is necessary to calculate In particular, it is easy to understand that control design becomes significantly more difficult as the number of links increases.

(3) Response characteristics can be easily adjusted. If well-adjusted, input calculations will become easier and the amount of calculations can be significantly reduced.

(4) The effect of reducing the amount of calculation becomes more significant in the case of a multi-link system.

(5) The control law of the present invention provides positioning control between points (
It can also be directly applied to Pa1nt to Pa1nt Control). Hereinafter, the effects of the present invention, including those described in the above items, will be explained in detail.

Now, since orbit control usually only takes a finite amount of time, response coordination is required rather than steady-state characteristics.

When using sliding mode control, a speed of convergence to the switching surface≦=0 is particularly required. this is(
It is related to the constant of 1 in equation (25), and if kl is set large, convergence is faster according to equation (13). When the state of the system enters the switching plane s = 0, the behavior is white, + c
, e , = O, and is expressed by n linear differential equations, so interference between each link is removed, making it look like an n ill independent single link system. When trying to quickly converge the error to zero, it is sufficient to simply increase C1.

Therefore, the application of the control law is very simple and highly practical.

Further, the control law of equation (25) is expressed by the nominal value of the physical parameter of the robot arm, the upper limit value of the deviation away from the nominal value, and the maximum value of the disturbance, and does not use the rX value at all.

In other words, if the physical parameter fluctuations of the robot arm and the disturbance values are within the limits, the steady-state characteristics of the system will not change, indicating that the system is robust against parameter fluctuations and disturbances. That is, since the control law is described using nominal values and limit values, it shows the simplicity of design. This is because although the identified values of the physical parameters are normally taken, the nominal values here can also be arbitrarily determined by the designer, which allows the creation of control laws with various characteristics. It is. When actually using the system, it is sufficient to create a control law that is easiest to apply depending on the specific control target and the usable control 9M setting.

Next, we will discuss reducing the amount of control calculations. The problem of the amount of calculation was one of the troublesome problems in the robot arm control section. That is, many operations must be performed within the control hyperperiod of l sampling.

Reducing the amount of calculation has long been a focus of attention, and various efforts have been made. However, in controlling a robot arm, there are many calculations of trigonometric functions (sine, cosine),
Calculating these in real time takes a lot of time. However, since this control method uses only the nominal values of the physical parameters and their upper limit values, it is possible to eliminate the calculation of these trigonometric functions by utilizing the boundedness of the trigonometric functions. for example,
Considering Coriolis and centrifugal force B (θ, θ), B (θ
, θ)=Σ f L)(s in θ, COS
θ) θ, θ.

is a time-varying coefficient f (s i n θ, COS θ), so it takes time to calculate it in real time, and the value is not very large. Here, if B' (θ, J) is set to zero, then B (θ, J) becomes B (θ, J) according to equation (14).
4) Since it becomes an upper bound f steel and the coefficient of the quadratic polynomial formula % θ) of 1θ, 1 becomes one constant, the calculation and amount can be significantly reduced. Other terms, such as gravity, can in principle be treated in the same way.

However, in this case, the value in brackets [ ] in equation (25) increases, and as the state approaches the bending plane, the discontinuous sign function sgn(s) causes the width of the discontinuous fluctuation of the input to become large. Therefore, it is undesirable to cause mechanical resonance to be excited.

In other words, although it is possible to easily control without using accurate parameters, the range of input fluctuations can be suppressed by using values closer to accurate physical parameters. When actually designing, it is necessary to consider the trade-off between the amount of calculation and the range of input variation.

[Brief explanation of the drawing]

Fig. 1 is a diagram showing the configuration when applying the control law of the present invention to a robot arm, Fig. 2 is a diagram for explaining the sign function sgn, , (s), and Fig. 3 is a diagram showing the saturation function sat (s /ε
), Figure 4 is a diagram to explain a thin layer that guarantees convergence of the system state, Figure 5 is a model of a two-link arm, and Figure 6 is a diagram showing the control law equation (25). Figure 7 shows the change in input during linear trajectory tracking when using the control law (25).
Fig. 8 is a stick diagram showing the state of linear trajectory tracking when using the control law equation (26), Fig. 9
The figure shows the change in input for linear trajectory tracking when using the control law equation (26), Figure 10 shows the linear trajectory tracking error when using the control law equation (26), and Figure 11 is the control law (2
5) A stick diagram showing how the circular trajectory is followed when formula is used, Figure 12 is a diagram showing the input change when following the circular trajectory when the control law formula (26) is used, and Figure 13 is the control Rule (26)
It is a figure which shows the circular orbit following error when using a formula. 1...Robot arm 2...Control law (25) formula % Formula % Wood's control law ε A late arrival at the end of the robot system Diagram 1 Diagram ■ Sign function sgn (S) Diagram for clearing the mini Figure 20 Figure 3 Figure 4 Figure 2 Model diagram of Rin-4 Figure 6 Figure 7 Figure 8 Figure 9 Figure 10 Figure 11 Figure 12 Figure 13

Claims (3)

[Claims] (1) In the trajectory control of the robot arm, input torque is calculated using nominal values, which are approximate values of the physical parameters of the robot arm, such as moment of inertia, link length, center of gravity, mass, and coefficient of viscous friction. The error between the nominal value and the true value of each physical parameter is set as the upper limit value, and this is used as the control amount. and a target joint angular velocity, the input torque is switched on a switching surface set on a plane formed by the deviation between the target joint angular velocity and the target joint angular velocity, and the deviation from the trajectory is corrected by the switching input torque. Method. (2) In the robot arm, the control law that gives the input torque to each joint is the nominal value of centrifugal force and Coriolis force, the nominal value of viscous friction force, the nominal value of gravity, the nominal value of the inertia matrix, and the coefficient of the switching surface. multiplied by the angular velocity error, the product of any positive constant and the state of the switching surface, and the inertia term i
The product of half of the sum of each upper bound of the row element and the state of the switching surface, the upper bound of disturbance, the upper bound of the product of inertia matrix and target angular acceleration, centrifugal force and Coriolis The upper bound of force, the upper bound of gravity, the upper bound of the inertia matrix multiplied by the switching surface coefficient and the absolute value of the angular velocity error, and the upper bound of the viscous friction coefficient multiplied by the absolute value of the angular velocity. Trajectory control of a robot arm according to claim 1, characterized in that the upper limit value of the viscous friction force and each added value are constituted by the sum of an amount that switches between positive and negative depending on the sign of the switching surface. Method. (3) Claim 2, characterized in that in the control law for providing the input torque, the sign function is a saturation function.
A method for controlling the trajectory of a robot arm as described in Section 1.