CN105974797B - Consider the wire saws parallel robot motion control method of elasticity effect and compensation - Google Patents
Consider the wire saws parallel robot motion control method of elasticity effect and compensation Download PDFInfo
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Abstract
The wire saws parallel robot motion control method for considering elasticity effect and compensation, is related to robot.System dynamics equation is built, for rope tension optimization and controller design；According to desired moving platform movement locus, it is based on system dynamics equation and constraints, the rope tension dynamic optimization model for being up to object function with rigidity weighting, optimization rope tension distribution is used to calculate deflection；Moving platform position and attitude error caused by traction rope flexible deformation is analyzed by forward kinematics solution according to practical rope length and deflection；Using the actual motion state of vision measurement moving platform, and using its deviation between desired motion state as controlled quentity controlled variable；Design controller；According to the controller of design, the instruction of driving force square is calculated, it is final to control moving platform movement locus and rope tension, meet duty requirements.
Description
Technical field
The present invention relates to robots, are transported more particularly, to the wire saws parallel robot of a kind of consideration elasticity effect and compensation
Flowing control method.
Background technology
Wire-driven parallel support system (Cable-Driven Parallel Supporting System, also referred to as Suo Binglian
Support system) it is a kind of novel mechanism based on robot technology, compared with traditional rigid connection parallel institution, the former is maximum
The advantages of be with scalability, by transmission mechanism adjust rope length be easily achieved larger working space, simultaneously
Have the characteristics that simple in structure, at low cost, inertia is small, movement is flexible, is highly suitable for mechanical processing, robot crane, aviation
The fields such as space flight, it has also become the hot spot of Recent study.
The substantially complicated close coupling of wire-driven parallel support system, multiple-input and multiple-output, nonlinear and time-varying system,
Dynamic analysis is the key that realize the mechanism kinematic with control.Further, since rope can only tension cannot be pressurized, it is desirable that system
Rope should be in tensioning state always when dynamic change, this feature is also determined must consider rope simultaneously when designing controller
Tension and motion pose, i.e. power/position mixing control.The application that some are required with moving platform high-precision motion, such as wire saws height
Fast video camera, wind tunnel test wire-driven parallel support system etc. also need the shadow for further considering rope elasticity to moving platform pose
It rings, and control compensation.
Compared with developing more mature rigid parallel robot control technology, the control of rope traction and parallel-connection mechanism at present is ground
Study carefully relatively fewer.In place of the control method of the two has similar use for reference, but also have any different.Especially with respect to power/position mixing control
Strategy, the former typically refers to end effector and contacts the power generated with external environment, may influence the TRAJECTORY CONTROL of pure position
Mode, therefore force detection sensor is introduced, it goes to correct desired location track using the power error of feedback, to reach control mesh
's；And the latter is primarily to ensure that drag rope is in tensioning state always in moving platform dynamic test, and meet certain
Rigidity requirement, while to model Pose Control, continuous real-time optimal control is carried out to rope tension.In addition, wire saws
The elasticity effect restricted in parallel robot is also to have to consider.
Currently, for the high-precision control problem of wire saws parallel robot, though there is document to carry out control emulation and tested
Card, such as：Foreign countries, Lamaury etc. are directed to ReelAx8 rope principle of parallel model machines, are based on driving spatial design PID controller, with
Motor corner is that feedback quantity controls rope lengths, adjustment moving platform pose (Lamaury J, Gouttefarde M, Michelin
M,et al.Design and control strategies of a redundant suspended cable-driven
parallel robot.In:Lenarcic J,Husty M(eds)Advances in robot
kinematics.Springer,Berlin,2012:237–244)；Bayani etc. is directed to plane wire saws parallel robot, adopts
With monocular vision measure and adaptive sliding-mode observer, but only kinematics control (Bayani H, Masouleh M T, Kalhor
A.An experimental study on the vision-based control and identification of
planar cable-driven parallel robots.Robotics and Autonomous Systems,2016,75:
187-202).The country, the half-closed loop control that enemy original hawk etc. to rope traction and parallel-connection mechanism based on rigidity enhance emulate (Liu
It is glad, Chou Yuanying, the rigidity enhancing and motion control of Sheng English wind tunnel test wire saws redundant parallel robots, aviation journal,
2009,30(6):1156-1164)；And it is directed to high speed wire saws cameras people, it is based on terminal position space, before devising PD
Feedback controller and interference observer are to realize stability contorting (Wei Huiling, Chou Yuanying, the Sheng English high speed wire saws video cameras of movement
Motion stabilization control [J] the Xian Electronics Science and Technology University journal of device people, 2016,05:70-77).But due to wire saws parallel machine
The application background of structure is different, and the critical issue that of interest and needs solve also is not quite similar, and the studies above is without reference to rope bullet
The influence of property, control accuracy can't fully meet requirement.
For wire saws parallel robot in high-precision applications such as wind tunnel test support system, mobile camera trailer systems
In the case of, it not only needs to consider the impact analysis of system stiffness and rope deformation to moving platform pose, design is also needed to mend
The force-location mix control method of corresponding error is repaid, it is therefore proposed that a kind of wire saws parallel machine considering rope elasticity effect and compensation
Device people's motion control method, is of great significance to its engineer application.
Invention content
The purpose of the invention is to overcome deficiencies of the prior art, one kind is provided and is mainly used to solve to restrict to lead
The problems such as drawing the existing control method of parallel robot existing kinematic accuracy be not high during the motion, system stiffness is relatively low
Consider the wire saws parallel robot motion control method of elasticity effect and compensation.
The present invention includes the following steps：
1) system dynamics equation is built, for rope tension optimization and controller design；
2) according to desired moving platform movement locus, it is based on system dynamics equation and constraints, uses and is added with rigidity
Power is up to the rope tension dynamic optimization model of object function, optimization rope tension distribution, and then calculates deflection；
3) moving platform caused by traction rope flexible deformation is analyzed by forward kinematics solution according to practical rope length and deflection
Position and attitude error；
4) the actual motion state of vision measurement moving platform is used, and will be between actual motion state and desired motion state
Deviation as controlled quentity controlled variable；
5) controller is designed, is specifically included：The proportion differential feedback controller based on position and attitude error is established, to meet movement
Required precision；Using rope tension dynamic optimization as feedforward controller, to ensure system stiffness, while rope being avoided to relax；Rope is added
Long variation correction term, error caused by compensation is elastic enhance the robustness of system, ensure that moving platform is moved along desired trajectory；
6) according to the controller of design, the instruction of driving force square is calculated, final moving platform movement locus and the rope of controlling is drawn
Power meets duty requirements.
In step 1), the system dynamics equation uses following expression：
System dynamics adoption status equation representation, x in formula1、x2、x3、x4Indicate state vector, wherein x1=X,x3=θ,X indicates that the pose vector of moving platform, θ are the corner vector of machine shaft.M0Motor is arrived to be equivalent
Moment of inertia matrix on axis；C0For equivalent viscous friction coefficient matrix；M (X) is the inertial matrix of moving platform；For
Non-linear Ge Shi centrifugal force matrix,For speed term；wgFor the gravity vector of moving platform；weFor external load suffered by moving platform, such as
Aerodynamic load；J is the Jacobian matrix of system；KsFor the extensional rigidity matrix of rope；L1、L2Correspond respectively to practical rope
Long and theoretical rope length；τ is motor driving force moment vector；R is the gearing factor of ball-screw, related with helical pitch；Indicate single order
Derivative；()-1Representing matrix it is inverse；()TThe transposition of representing matrix.
In step 2), the rope tension dynamic optimization model uses following expression：
To improve system principal direction rigidity as object function, optimization rope tension is distributed model；F () is optimization aim letter in formula
Number, T are that rope is pulling force vector, and λ is Lagrange multiplier；Kj,jFor j-th of diagonal entry of stiffness matrix, wjIt is weighted for j-th
Coefficient, j=1...6；tiThe pulling force restricted for i-th, i=1...8；tmin、tmaxFor rope tension lower limiting value and upper limit value；J be
The Jacobian matrix of system；M (X) is the inertial matrix of moving platform；For non-linear Ge Shi centrifugal force matrix；X is moving platform
Pose vector；For speed term；For acceleration item；wgFor the gravity vector of moving platform；weIt is carried to be external suffered by moving platform
Lotus；()TThe transposition of representing matrix；Min expressions are minimized；∑ indicates summation algorithm；| | expression takes absolute value；()2Expression takes
Square.
In step 5), the controller uses following expression：
In formula, τ torques in order to control；R is the gearing factor of ball-screw；(JT)+Indicate the puppet after Jacobian matrix transposition
It is inverse；KpIt is the ratio control gain of system；KdIt is the differential control gain of system；TdIt is the rope tension obtained by Inverse Dynamics
Feedforward term；K' is elastic compensating item proportional gain；XdIt is the desired trajectory of moving platform；X is the actual motion track of moving platform；L1、
L2Correspond respectively to practical rope length and theoretical rope length；Indicate first derivative.Designed controller includes three, wherein first
Item is moving platform Pose Control feedback term, and Section 2 is that rope tension controls feedforward term, and Section 3 is elastic compensating item of restricting.
The present invention is directed to the motion control of wire saws parallel robot moving platform, proposes to consider rope elasticity effect and compensation
High-accuracy control method has the following advantages：Based on stiffness optimization criterion, quantitative analysis rope flexible deformation is to moving platform pose
It influences, not only contributes to improve system stiffness, more moving platform high-precision motion control provides support；The design of controller was both wrapped
The PD feedback terms based on pose are included, and include rope tension feedforward term, and rope elastic compensating item, may finally realize wire saws simultaneously
Join the high-precision control of robot moving platform；The analysis can be widely applied to the movement of wire saws parallel robot with control method
In control.
Description of the drawings
Fig. 1 is a kind of wire saws parallel robot motion control method principle frame of compensation rope elasticity effect of the present invention
Figure.
Fig. 2 is that a kind of consideration rope elasticity effect of the present invention and the wire saws parallel robot motion control method of compensation are set
Count flow chart.
Fig. 3 is a kind of typical eight wire saws six degree of freedom redundant constaint parallel robot.
Fig. 4 represents the rope tension optimum results based on rigidity in the case where having method.
Fig. 5 is to restrict caused by flexible deformation along X-axis, Y-axis, the site error of Z axis in the case where having method.
Fig. 6 is the attitude errors such as roll angle, pitch angle and yaw angle caused by rope flexible deformation in the case where having method.
Fig. 7 is the tracking error of the pitch angle under control system of the present invention.
Specific implementation mode
Below in conjunction with drawings and the specific embodiments, invention is further described in detail
Fig. 1 show the wire saws parallel robot motion control side of a kind of the consideration rope elasticity effect and compensation of the present invention
Method functional block diagram, Fig. 2 show the wire saws parallel robot movement control of a kind of the consideration rope elasticity effect and compensation of the present invention
Method design flow diagram processed.System dynamics equation is built first, is optimized for rope tension；According to desired motion track, it is based on
System dynamics equation and rope tension constraints are up to object function with rigidity weighting, establish rope tension dynamic optimization mould
Type；Moving platform position and attitude error caused by traction rope flexible deformation is analyzed by forward kinematics solution according to practical rope length；By regarding
Feel the pose for measuring moving platform, it would be desirable to which the error between movement locus and actual motion track is as regulation and control amount；Design is based on
The PD feedback controllers of position and attitude error, to meet kinematic accuracy requirement；Using rope tension dynamic optimization as feedforward controller, to protect
System stiffness is demonstrate,proved, while rope being avoided to relax；Long change of rope correction term is added, error caused by compensation is elastic enhances the Shandong of system
Stick ensures that moving platform is moved along desired trajectory.It is as follows：
1) system dynamics equation is built, is optimized for rope tension.For the system dynamics of wire saws parallel robot
Equation is represented by：
System dynamics adoption status equation representation, x in formula1、x2、x3、x4Indicate state vector, wherein x1=X,x3=θ,X indicates that the pose vector of moving platform, θ are the corner vector of machine shaft.M0Motor is arrived to be equivalent
Moment of inertia matrix on axis；C0For equivalent viscous friction coefficient matrix；M (X) is the inertial matrix of moving platform；For
Non-linear Ge Shi centrifugal force matrix,For speed term；wgFor the gravity vector of moving platform；weFor external load suffered by moving platform, such as
Aerodynamic load；J is the Jacobian matrix of system；KsFor the extensional rigidity matrix of rope；L1、L2Correspond respectively to practical rope
Long and theoretical rope length；τ is motor driving force moment vector；R is the gearing factor of ball-screw, related with helical pitch；Indicate single order
Derivative；()-1Representing matrix it is inverse；()TThe transposition of representing matrix.
2) according to desired moving platform motion state, it is based on system dynamics equation and rope tension constraints, with rigidity
Weighting is up to target, the distribution of dynamic optimization rope tension.Wherein Optimized model is represented by：
To improve system stiffness as object function, optimization rope tension is distributed model；F () is optimization object function in formula, and T is
Rope system pulling force vector, λ are Lagrange multiplier；Kj,jFor j-th of diagonal entry of stiffness matrix, wjFor j-th of weighting coefficient, j
=1...6；tiThe pulling force restricted for i-th, i=1...8；tmin、tmaxFor rope tension lower limiting value and upper limit value；J is the refined of system
Gram compare matrix；M (X) is the inertial matrix of moving platform；For non-linear Ge Shi centrifugal force matrix；X is the pose of moving platform
Vector；For speed term；For acceleration item；wgFor the gravity vector of moving platform；weFor external load suffered by moving platform；()TTable
Show the transposition of matrix；Min expressions are minimized；∑ indicates summation algorithm；| | expression takes absolute value；()2Expression is squared.
3) according to rope tension optimum results, rope deflection is calculated：
In formula, Δ LiFor i-th long change of rope amount；tiFor rope tension real-time optimization value；t0For initial pretightening force；S is rope
Cross-sectional area；Y is rope elasticity modulus；LiFor i-th rope length theoretical value.
Further the attained pose of moving platform is obtained, is done with expected pose using Numerical Iteration Method according to practical rope length
Difference, you can obtain moving platform position and attitude error caused by rope length deformation.
4) use vision measurement moving platform actual motion state, and using its deviation between desired motion state as
Motion control amount；
δ=Xd-X (4)
In formula, δ is error；XdFor expected pose；X is the attained pose of vision measurement.
5) according to system dynamics equation, a kind of wire saws parallel robot motion control of compensation rope flexible deformation is designed
Device.Specifically it is expressed as follows：
(a) the PD control device fed back based on moving platform pose is designed, to meet kinematic accuracy requirement：
In formula, τ1For first item control moment；R is the gearing factor of ball-screw；(JT)+Indicate Jacobian matrix transposition
Pseudoinverse afterwards；KpIt is the ratio control gain of system；KdIt is the differential control gain of system；XdIt is the desired trajectory of moving platform；X
It is the actual motion track of moving platform；Indicate first derivative.
(b) rope tension feedforward controller is designed, to meet fortune rigidity requirement：
τ2=rTd (6)
In formula, τ2For Section 2 control moment；R is the gearing factor of ball-screw；TdIt is the rope obtained by Inverse Dynamics
Pulling force feedforward term；(JT)+Indicate the pseudoinverse after Jacobian matrix transposition；M (X) is the inertial matrix of moving platform；It is non-
Linear Ge Shi centrifugal force matrixes；X is the pose vector of moving platform；For speed term；For acceleration item；wgFor the weight of moving platform
Force vector；weFor external load suffered by moving platform.
(c) long change of rope correction term is designed, error caused by flexible deformation is compensated：
τ3=K'(L1-L2) (8)
In formula, τ3For Section 3 control moment；K' is elastic compensating item proportional gain；L1、L2Correspond respectively to practical rope length
With theoretical rope length.
Then total control moment is represented by：
τ=τ1+τ2+τ3 (9)
In formula, τ is total control moment.
6) according to the controller of design, the instruction of driving force square is calculated, final moving platform movement locus and the rope of controlling is drawn
Power meets duty requirements.
Embodiment
By the wire saws parallel robot motion control method application of a kind of the consideration rope elasticity effect and compensation that are proposed
With the six-degree-of-freedom parallel robot of eight wire saws, the parallel robot as shown in figure 3, moving platform 1 by taking dummy vehicle as an example,
It is drawn by eight traction ropes 2；Pulley 3 is fixed in rack 4；Traction rope 2 adjusts length by motor by pulley 3, to change
The position of dummy vehicle 1 and posture；The pose of dummy vehicle 1 can be obtained by the measurement of monocular vision 5, be mounted on and be fixed on
On the holder 6 of rack 7.The present embodiment control method is implemented as follows：
1) 1 error analysis of moving platform caused by 2 flexible deformation of traction rope
It is defined according to coordinate system, quiet coordinate origin is located at the center of 4 bottom surface of rack, and three reference axis are orthogonal, and meets
The right-hand rule；Moving coordinate system is located on the barycenter of moving platform 1, and wherein X-axis is before moving platform axis direction, and Y-axis is along spanwise
Outside being directed toward, under Z axis is directed toward, and meet the right-hand rule with X-axis, Y-axis.When moving platform 1 is in zero-bit appearance, quiet coordinate system is sat with dynamic
Mark system is parallel.1 barycenter of moving platform at this time, that is, it is that (0,0, -582) mm is expressed as in quiet coordinate system to refer to point coordinates；Traction rope 2
With the tie point B of rack 4iIt indicates, i=1 ... 8, wherein in quiet coordinate system：B1(472,814.5, -1285)T, B2
(514.5, -772, -1286)T, B3(- 472, -814.5, -1286)T, B4(- 472, -814.5, -1286)T, B5(- 472,
814.5, -90)T, B6(514.5,772, -91)T, B7(472, -814.5, -91)T, B8(- 514.5, -772, -90)T, unit is
Mm, ()TIndicate transposition；The tie point P of traction rope 2 and moving platform 1iIt indicates, i=1 ... 8, wherein in moving coordinate system：P1
(30,19.1, -19.1)T, P2(30, -19.1, -19.1)T, P3(- 165, -26,0)T, P4(- 165,26,0)T, P5(- 165,26,
0)T, P6(30,19.1,19.1)T, P7(30, -19.1,19.1)T, P8(- 165, -26,0)T, unit mm, ()TIndicate transposition；
1 mass of moving platform is 1.06kg；In original state, according to standing balance, setting 2 initial pretightening force minimum value of traction rope is 10N,
And it is selected as Kevlar rope, elasticity modulus 43.9GPa.According to rope material property and moving platform quality, rope tension bound point is taken
It Wei not 35N and 300N.
By taking pitching angle theta sinusoidal motion as an example, i.e. θ=π/6*sin (t), simulation time t is set as moving platform desired trajectory
10s.Rope tension optimization is carried out first, and rigidity weighting coefficient is selected as (0.2,0.1,0.3,0.2,0.1,0.1) successively.According to rope
Pulling force result and deflection carry out Kinematic Problem solution using practical rope length, obtain the attained pose of moving platform 1；Into
And subtract each other with desired trajectory, moving platform position and attitude error caused by rope length deformation can be obtained.
2) control method of compensation rope flexible deformation
The rated output torque τ of driving motorr=0.64Nm, the equivalent rotary inertia to motor side of drive system are m0
=7.76 × 10-5kg·m2, the equivalent viscous damping coefficient to motor side is c0=2 × 10-3N·m·s；Transmission ball wire
The gearing factor of thick stick is r=a/2 π, and ball-screw screw pitch is a=0.005m.
Likewise, it would be desirable to which track is by taking moving platform pitch angle sinusoidal motion as an example.It is as follows to design controller：
Wherein, τ is total control moment；R is the gearing factor of ball-screw；(JT)+After indicating Jacobian matrix transposition
Pseudoinverse；It is adapted in PD feedback control items, proportional gain factor Kp=diag (0,0,0,0,3.5,0), differential gain COEFFICIENT Kd
=diag (0,0,0,0,450,0)；K'=50 × diag (1,1,1,1,1,1) in rope elastic compensating item；Diag () indicate with to
The diagonal matrix of element composition in amount；TdIt is the rope tension feedforward term obtained by Inverse Dynamics；The desired trajectory X of moving platformd
=[0,0,0,0, π/6*sin (t), 0]T, []TIndicate transposition；X is the actual motion track for the moving platform that vision measurement obtains；Indicate first derivative；L1、L2Correspond respectively to practical rope length and theoretical rope length.
3) it needs to carry out control compensation known to error result Fig. 5-Fig. 6, to improve kinematic accuracy.It is controlled aforementioned
Torque substitutes into system dynamics equation (1), may finally realize the high-precision motion of wire saws parallel robot moving platform 1.
Result by the control method of this embodiment is as shown in Figure 4 to 7.
Fig. 4 represents the rope tension optimum results based on rigidity in the case where having method；Fig. 5 represents elasticity of restricting in the case where having method
Three shaft position errors caused by deformation；Fig. 6 represents three attitude errors caused by rope flexible deformation in the case where having method；Fig. 7
Represent the tracking error of the pitch angle under control system of the present invention.Abscissa indicates run duration in all figures.Become by elasticity
Error result caused by shape needs to carry out control compensation it is found that improve moving platform kinematic accuracy.A kind of benefit proposed by the present invention
Tracking error can be effectively reduced by repaying the wire saws parallel robot motion control method of rope flexible deformation, improve wire saws simultaneously
Join the movenent performance of robot.
The present invention initially sets up the system dynamics equation comprising driving motor and moving platform, and wherein rope tension is expressed as becoming
The linear function of shape amount；It is instructed for given moving platform motion state, the rope tension dynamic optimization model weighted using rigidity,
Calculate rope length deflection；Kinematic Problem is further solved, using numerical method, analyzes and is moved caused by traction rope flexible deformation
Platform's position and pose error；Controller is designed according to system dynamics equation, using the pose of vision measurement moving platform, as directly anti-
Feedback；Design proportion Derivative Feedback and rope tension feedforward controller, while long change of rope correction term is added, it compensates caused by elasticity accidentally
Difference enhances the robustness of system, ensures that moving platform is moved along desired trajectory；According to designed controller, control driving is calculated
Torque command, final realize control the high precision position and posture of moving platform with rope tension.
Claims (2)
1. considering the wire saws parallel robot motion control method of elasticity effect and compensation, it is characterised in that including walking as follows
Suddenly：
1) system dynamics equation is built, for rope tension optimization and controller design；
The system dynamics equation uses following expression：
System dynamics adoption status equation representation, x in formula1、x2、x3、x4Indicate state vector, wherein x1=X,
x3=θ,X indicates that the pose vector of moving platform, θ are the corner vector of machine shaft；M0To be equivalent on motor shaft
Moment of inertia matrix；C0For equivalent viscous friction coefficient matrix；M (X) is the inertial matrix of moving platform；For non-linear brother
Family name's centrifugal force matrix,For speed term；wgFor the gravity vector of moving platform；weFor external load suffered by moving platform, such as air force
Load；J is the Jacobian matrix of system；KsFor the extensional rigidity matrix of rope；L1、L2Correspond respectively to practical rope length and theory
Rope length；τ is motor driving force moment vector；R is the gearing factor of ball-screw, related with helical pitch；()·Indicate first derivative；
()-1Representing matrix it is inverse；()TThe transposition of representing matrix；
2) according to desired moving platform movement locus, it is based on system dynamics equation and constraints, is used with rigidity weighting most
The greatly rope tension dynamic optimization model of object function, optimization rope tension distribution, and then calculate deflection；
The rope tension dynamic optimization model uses following expression：
To improve system principal direction rigidity as object function, optimization rope tension is distributed model；F () is optimization object function, T in formula
It is pulling force vector for rope, λ is Lagrange multiplier；Kj,jFor j-th of diagonal entry of stiffness matrix, wjFor j-th of weighting system
Number, j=1...6；tiThe pulling force restricted for i-th, i=1...8；tmin、tmaxFor rope tension lower limiting value and upper limit value；J is system
Jacobian matrix；M (X) is the inertial matrix of moving platform；For non-linear Ge Shi centrifugal force matrix；X is moving platform
Pose vector；For speed term；For acceleration item；wgFor the gravity vector of moving platform；weFor external load suffered by moving platform；
()TThe transposition of representing matrix；Min expressions are minimized；∑ indicates summation algorithm；| | expression takes absolute value；()2Expression is made even
Side；
3) moving platform pose caused by traction rope flexible deformation is analyzed by forward kinematics solution according to practical rope length and deflection
Error；
4) the actual motion state of vision measurement moving platform is used, and will be inclined between actual motion state and desired motion state
Difference is used as controlled quentity controlled variable；
5) controller is designed, is specifically included：Establish the proportion differential feedback controller based on position and attitude error；Rope tension dynamic is excellent
It is turned to feedforward controller；Long change of rope correction term is added, error caused by compensation is elastic enhances the robustness of system, ensures
Moving platform is moved along desired trajectory；
6) according to the controller of design, the instruction of driving force square is calculated, it is final to control moving platform movement locus and rope tension, completely
Sufficient duty requirements.
2. considering the wire saws parallel robot motion control method of elasticity effect and compensation, feature as described in claim 1
It is in step 5), the controller uses following expression：
In formula, τ is motor driving force moment vector；R is the gearing factor of ball-screw；(JT)+After indicating Jacobian matrix transposition
Pseudoinverse；KpIt is the ratio control gain of system；KdIt is the differential control gain of system；TdIt is to be drawn by the rope that Inverse Dynamics obtain
Power feedforward term；K' is elastic compensating item proportional gain；XdIt is the desired trajectory of moving platform；X indicates the pose vector of moving platform；L1、
L2Correspond respectively to practical rope length and theoretical rope length；()·Indicate first derivative；Designed controller includes three, wherein the
One is moving platform Pose Control feedback term, and Section 2 is that rope tension controls feedforward term, and Section 3 is rope elastic compensating item.
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