WO2020259132A1 - 绳驱动联动式机械臂的动力学建模及其绳索张力优化方法 - Google Patents

绳驱动联动式机械臂的动力学建模及其绳索张力优化方法 Download PDF

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WO2020259132A1
WO2020259132A1 PCT/CN2020/091146 CN2020091146W WO2020259132A1 WO 2020259132 A1 WO2020259132 A1 WO 2020259132A1 CN 2020091146 W CN2020091146 W CN 2020091146W WO 2020259132 A1 WO2020259132 A1 WO 2020259132A1
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rope
joint
linkage
tension
driving
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PCT/CN2020/091146
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English (en)
French (fr)
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孟得山
梁斌
王学谦
李亚南
刘宇
王天舒
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清华大学深圳国际研究生院
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Publication of WO2020259132A1 publication Critical patent/WO2020259132A1/zh

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    • B—PERFORMING OPERATIONS; TRANSPORTING
    • B25—HAND TOOLS; PORTABLE POWER-DRIVEN TOOLS; MANIPULATORS
    • B25J—MANIPULATORS; CHAMBERS PROVIDED WITH MANIPULATION DEVICES
    • B25J9/00—Programme-controlled manipulators
    • B25J9/16—Programme controls
    • B—PERFORMING OPERATIONS; TRANSPORTING
    • B25—HAND TOOLS; PORTABLE POWER-DRIVEN TOOLS; MANIPULATORS
    • B25J—MANIPULATORS; CHAMBERS PROVIDED WITH MANIPULATION DEVICES
    • B25J9/00—Programme-controlled manipulators
    • B25J9/16—Programme controls
    • B25J9/1656—Programme controls characterised by programming, planning systems for manipulators
    • B25J9/1664—Programme controls characterised by programming, planning systems for manipulators characterised by motion, path, trajectory planning

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  • the invention relates to the field of rope-driven linkage mechanical arms, in particular to a dynamic modeling method of a rope-driven linkage mechanical arm and a rope tension optimization method.
  • the rope-driven linkage manipulator has two control modes: one is the tension control mode, where a force sensor is installed on the rope as feedback to directly control the tension on the rope; the other is the rope length control mode, which treats the rope as an axial direction
  • the linear spring adjusts the rope tension indirectly by adjusting the rope driving law. Either way, the rope tension ultimately acts on the connecting rod and drives the robot to move.
  • the rope tension is limited by its own strength characteristics, mainly the rope material and radius, and the way the rope end is fixed.
  • rope slack is also a common problem. The rope elasticity reduces the end accuracy of the robot arm. All these problems restrict the application of rope-driven manipulators to a certain extent.
  • rope drive length planning often starts from the kinematics level without considering the dynamic characteristics of the manipulator.
  • the rope elasticity plays an important role in the dynamics of the manipulator.
  • the moving part of the rope-driven manipulator is often light in weight and small inertia. This feature makes the rope-driven manipulator have a high-speed response
  • the elasticity of the rope may make the system unstable and reduce the end positioning accuracy. Therefore, rope elasticity should be considered to fully tap the potential of rope-driven robots.
  • the present invention proposes a dynamic modeling method of a rope-driven linkage manipulator and a rope tension optimization method thereof, which realizes the optimization of the rope tension during the movement of the manipulator, and can be used for the rope-driven manipulator. Trajectory planning, vibration control and other aspects.
  • the invention discloses a dynamic modeling method of a rope-driven linkage mechanical arm, which includes:
  • the rope-driven linkage manipulator is equivalent to a rigid body part, two drive ropes and four linkage ropes, wherein the rigid body part includes three rotary joints and three connecting rods, and the three connecting rods pass through the The rotary joints are connected to each other; one end of the two driving ropes is respectively fixed on the connecting rod at the end, and the other end passes through each of the connecting rods in turn and then is controlled by the driver; the four linkage ropes are respectively wound in pairs Pass through the adjacent rotary joints to be cross-connected to the connecting rods;
  • q is the joint angle
  • M(q) is the generalized mass matrix
  • ⁇ is the joint moment
  • ⁇ c is the equivalent joint moment of the linkage rope mapped to the rotary joint.
  • the expression of the joint moment ⁇ is:
  • H(q) is the mapping relationship from the tension of the driving rope to the joint torque, and T is the tension of the driving rope;
  • E a is the Young's modulus of the driving rope
  • a a is the cross-sectional area of the driving rope
  • l a (q) is the length of the driving rope at the current moment
  • ⁇ l a is the total deformation of the driving rope
  • l a0 is the length of the driving rope at the initial moment
  • l am is the length of the driving rope driven by the driver.
  • the expression of the equivalent joint moment ⁇ c where the linkage rope is mapped to the rotary joint is:
  • k c is the coefficient of elasticity of the linkage rope
  • r is the winding radius of the linkage rope
  • ⁇ q is the angle difference between adjacent rotating joints
  • E c is the Young's modulus of the linkage rope
  • a c is the cross-sectional area of the linkage rope
  • l c0 is the length of the linkage rope.
  • the invention also discloses a method for optimizing the tension of the driving rope of the rope-driven linkage manipulator, which includes: using a global interpolation polynomial to estimate the state variable and the control variable, and deriving the polynomial to obtain the derivative of the state variable to discretize the differential equation
  • the optimal control problem is transformed into a nonlinear constraint problem, where the polynomial refers to the dynamic equation of the rope-driven linkage manipulator established in the above-mentioned dynamic modeling method, and the obtained nonlinear constraint problem includes dynamics Learning constraint requirements.
  • the optimal solution of the first minimization objective function is sought within the first nonlinear constraint requirement to determine the driving rope of the rope-driven linkage manipulator. Perform tension optimization.
  • the first nonlinear constraint requirement includes a dynamic constraint requirement
  • the expression of the dynamic constraint requirement is:
  • ⁇ H(q)u
  • q is the joint angle
  • ⁇ is the joint moment
  • ⁇ c is the equivalent joint moment of the linkage rope mapped to the joint
  • M is the generalized mass matrix
  • H(q) is the mapping relationship from the tension of the driving rope to the joint moment
  • t represents the time
  • t 0 represents the initial time
  • t f represents the end time
  • the first nonlinear constraint condition further includes boundary condition constraint requirements and inequality path constraint requirements; wherein the expression of the boundary condition constraint requirements is: ⁇ min ⁇ ⁇ (x(t 0 ), t 0 , x(t f ),t f ) ⁇ max , ⁇ is the boundary condition constraint parameter; the expression required by the inequality path constraint is: C min ⁇ C(x(t),u(t),t) ⁇ C max , C is the inequality Path constraint parameters.
  • the tension of the drive rope of the rope-driven linked mechanical arm can be optimized when the second nonlinear linear constraint requirement is satisfied.
  • the second nonlinear constraint requirement includes a dynamic constraint requirement, and the expression of the dynamic constraint requirement is:
  • ⁇ H(q)T
  • q is the joint angle
  • ⁇ is the joint moment
  • ⁇ c is the equivalent joint moment of the linkage rope mapped to the joint
  • M is the generalized mass matrix
  • H(q) is the mapping relationship from the tension of the driving rope to the joint moment
  • t represents the time
  • t 0 represents the initial time
  • t f represents the end time
  • T represents the driving rope Tension
  • E a is the Young's modulus of the driving rope
  • a a is the cross-sectional area of the
  • the second non-constrained condition further includes boundary condition constraint requirements and inequality path constraint requirements; wherein the expression of the boundary condition constraint requirements is: ⁇ min ⁇ ⁇ (x(t 0 ),t 0 ,x(t f ),t f ) ⁇ max , ⁇ is the boundary condition constraint parameter; the expression required by the inequality path constraint is: T min ⁇ T ⁇ T max , T represents the tension of the driving rope.
  • the beneficial effect of the present invention is that the dynamic modeling method of the rope-driven linkage manipulator proposed by the present invention comprehensively considers the elasticity of the driving rope, so that the end position error caused by the deformation of the driving rope is compensated . Furthermore, the method for optimizing the tension of the driving rope of the rope-driven linkage manipulator provided by the present invention comprehensively considers the dynamics of the manipulator to make the manipulator system more stable and ensure that the rope is always under tension during the movement.
  • the tension optimization method for the driving rope of the robotic arm helps to select the appropriate rope and guide the mechanism design; second, through optimization to reduce the maximum tension of the rope during the movement , It can reduce the possibility of rope breaking; third, the rope always maintains a certain tension to prevent slack during the movement; fourth, the end position error caused by the elasticity of the rope is compensated.
  • Figure 1a is a diagram of the straightened state of the rope-driven linkage type mechanical arm of the preferred embodiment of the present invention
  • Fig. 1b is a diagram of the equal curvature bending state of the mechanical arm in Fig. 1a;
  • Figure 2 is a schematic diagram of the linkage rope of the robotic arm in Figure 1a;
  • Figure 3a and Figure 3b are the tension of the two driving ropes of the robot arm in rope tension control mode
  • Figures 4a and 4b show the joint angle and joint angular velocity of the manipulator in rope tension control mode, respectively;
  • Figures 5a and 5b are the driving lengths of the two driving ropes of the robot arm in rope length control mode
  • Figures 6a and 6b are the joint angle and joint angular velocity of the manipulator in the rope length control mode, respectively;
  • Figures 7a and 7b show the tension of the two driving ropes of the robot arm in the rope length control mode.
  • the manipulator is composed of three parts, namely a rigid body part 10, two drive ropes 20 and four linkage ropes 30. .
  • the rigid body part 10 consists of three rotary joints 11 (including a first rotary joint 111, a second rotary joint 112, and a third rotary joint 113) and three links 12 (including a first link 121, a second link 122, and a Three connecting rods 123) are composed, three connecting rods 12 are connected to each other through a rotary joint 11 in turn, and the connecting rods 12 move in a plane.
  • the mechanical arm is actively driven by two independently driven drive ropes 20.
  • the discs on both sides of the connecting rod 12 have rope holes.
  • the driving rope 20 passes through the rope holes, and the end is fixed on the end connecting rod 12 through the rope head.
  • One end of the two driving ropes 20 are respectively fixed on the connecting rod 12 at the end, and the other end is driven by the motor after passing through each connecting rod 12 in turn.
  • the driving rope 20 is tightened by the motor, and the driving rope 20 generates tension. A force is generated on the connecting rod 12 to drive the connecting rod 12 to move.
  • the four linkage ropes 30 respectively bypass the adjacent rotating joints 11 in pairs to be cross-connected to the connecting rod 12 (that is, the four linkage ropes 30 are divided into two groups of two linkage ropes, and the two linkage ropes of one group are
  • the rope 301 respectively bypasses the first rotary joint 111 and the second rotary joint 112 to be cross-arranged on the first link 121 between the first rotary joint 111 and the second rotary joint 112, wherein the two linked ropes 301
  • the two ends are respectively fixedly connected to the base 40 and the second link 122;
  • the two linkage ropes 302 of the other group respectively bypass the second revolute joint 112 and the third revolute joint 113 to be arranged in the second revolute joint 112 crosswise
  • the second link 122 between the third revolving joint 113, wherein both ends of the two linkage ropes 302 are fixedly connected to the first link 121 and the third link 123), so that all the revolving joints 11 have
  • q is the joint angle
  • M(q) is the generalized mass matrix
  • ⁇ is the joint moment
  • ⁇ c is the equivalent joint moment of the linkage rope mapped to the rotary joint.
  • FIG. 2 it is a schematic diagram of the linkage rope 30.
  • the winding radius of the linkage rope 30 is r.
  • the pre-tightening force of the rope 30 is T c0 , then:
  • T c1 and T c2 are the tensions of the two linkage ropes 30 respectively
  • ⁇ c is the equivalent joint moment of the linkage rope mapped to the rotary joint
  • k c is the elastic coefficient of the linkage rope 30
  • E c is the Young's modulus of the linkage rope 30
  • a c is the cross-sectional area of the linkage rope 30
  • l c0 is the length of the linkage rope 30.
  • the motor pulls the driving rope 20, and the tension of the driving rope 20 changes accordingly.
  • the law of tension change depends on the stretching law of the driving rope 20:
  • H(q) is the mapping relationship between the tension of the driving rope 20 and the joint torque
  • T is the tension of the driving rope 20
  • ⁇ is the joint torque
  • E a is the Young's modulus of the driving rope 20
  • a a is the driving rope 20
  • L a (q) is the length of the drive rope 20 at the current moment
  • ⁇ l a is the total deformation of the drive rope 20
  • l a0 is the length of the drive rope 20 at the initial moment
  • l am is the length of the drive rope 20 driven by the motor
  • the length is the length of the driving rope 20 pulled by the motor.
  • the following is combined with the dynamic equation of the rope-driven linkage manipulator to optimize the tension of the driving rope of the rope-driven linkage manipulator.
  • the tension optimization of the driving rope is essentially an optimal control problem.
  • the Matlab optimization tool can be used Box GPOPS is optimized. Specifically: the global interpolation polynomial is used to estimate the state variable and the control variable, the derivative of the state variable is obtained from the polynomial to discretize the differential equation, the optimal control problem is transformed into a nonlinear constraint problem, and the calculation efficiency is greatly improved.
  • the polynomial refers to the dynamic equation of the rope-driven linkage manipulator established above, and the nonlinear constraint problems obtained include dynamic constraint requirements.
  • ⁇ is the function including the state quantity of the boundary condition
  • g is the integral term function including the state quantity of the whole process
  • t is the time
  • t 0 is the initial time
  • t f is the end time
  • x is the state variable of the system
  • u is the system Control variables.
  • the dynamic constraints (dynamic equations), boundary condition constraints (the range of state variables at the initial time and the end time) and the inequality path constraints (the range of state variables and control variables during motion) that need to be satisfied are:
  • ⁇ is the boundary condition constraint parameter
  • C is the inequality path constraint parameter
  • the tension of the driving rope is used as the control input of the system.
  • the objective function contains two parts. ⁇ is the function including the state quantity of the boundary conditions, and g is the The integral term function of the state quantity of the whole process; at this time, the objective function only considers the second term and is defined as the maximum tension of the driving rope, that is, the first minimized objective function is
  • the first unconstrained requirement includes dynamic constraint requirements, boundary condition constraint requirements, and inequality path constraint requirements. Combined with the dynamic constraint condition expression in the first line of equation (5), the dynamic constraint requirement in the tension control mode is Need to meet:
  • the state variable x and the control variable u are respectively:
  • q is the joint angle
  • Is the joint angular velocity
  • Is the Coriolis force and the centripetal force term
  • ⁇ is the joint moment
  • ⁇ c is the equivalent joint moment of the linkage rope mapped to the joint
  • M is the generalized mass matrix
  • H(q) is the mapping relationship from the tension of the driving rope to the joint moment
  • T 1 and T 2 respectively represent the tension of the two driving ropes
  • q 1 , q 2 , and q 3 respectively represent the joint angles of the three rotary joints.
  • boundary condition constraint requirements and the inequality path constraint requirements are respectively the constraint expressions in the second and third lines in equation (5).
  • the tension of the drive rope of the rope-driven linkage manipulator can be optimized when the second nonlinear constraint requirement is satisfied.
  • the rope length is used as the system control input.
  • the rope length can be obtained through the corresponding relationship with the motor encoder.
  • the inequality path constraint is used to limit the tension range of the driving rope.
  • the second non-constrained requirements include dynamic constraint requirements, boundary condition constraint requirements and inequality path constraint requirements, which combine the dynamic constraint condition expression in the first line of equation (5), and the dynamic constraint requirement in the rope length control mode Need to meet:
  • the state variable x and the control variable u are respectively:
  • q is the joint angle
  • Is the joint angular velocity
  • Is the Coriolis force and the centripetal force term
  • ⁇ is the joint moment
  • ⁇ c is the equivalent joint moment of the linkage rope mapped to the joint
  • M is the generalized mass matrix
  • H(q) is the mapping relationship from the tension of the driving rope to the joint moment
  • l 1 and l 2 respectively represent the length of the two driving ropes driven by the motor
  • q 1 , q 2 , and q 3 respectively represent the joint angles of the three rotary joints
  • T represents the tension of the driving rope
  • E a is the Young's of the driving rope Modulus
  • a a is the cross-sectional area of the driving rope
  • l a (q) is the length of the driving rope at the current moment
  • l a0 is the length of the driving rope at the initial moment.
  • the boundary condition constraint requirement is the constraint expression in the second line of equation (5).
  • T represents the tension of the driving rope
  • T min is determined by the minimum tension of the driving rope without slack
  • T max is determined by the maximum tension that the rope can withstand.
  • the mechanical arm is driven by two driving ropes 20, and the three rotary joints 11 are linked by four linkage ropes 30 to achieve equal curvature bending of the entire arm, and the connecting rod 12 moves in a plane.
  • zeros(1,6) represents a matrix of 0 with one row and six columns
  • ones(1,3) represents a matrix of 1 with one row and three columns
  • x(t 0 ) and x(t f ) are the states at the initial time and end time, respectively Variable value
  • x min and x max are the minimum and maximum values of the state variables
  • u min and u max are the minimum and maximum values of the control variables
  • u(t 0 ) guess and u(t f ) guess are respectively
  • the guess value of the input at the initial time and the end time are some initial conditions set.
  • Figure 3a and Figure 3b it is the optimized control variable input of the robot arm, that is, the tension of the two driving ropes.
  • the maximum value of the tension of the two ropes is 0.22N and the minimum value is 0.05N;
  • Figure 4a and Figure 4b shows the optimized state variables of the robotic arm.
  • Figure 4a shows the obtained joint angles.
  • the joint angles at the end time are both -5°.
  • Figure 4b shows the obtained joint angular speeds.
  • the joint angular velocity at the end time is 0, which satisfies the constraints at the end time. .
  • zeros(1,6) represents a matrix of 0 with one row and six columns
  • ones(1,3) represents a matrix of 1 with one row and three columns
  • x(t 0 ) and x(t f ) are the states at the initial time and end time, respectively Variable value
  • x min and x max are the minimum and maximum values of the state variables
  • u min and u max are the minimum and maximum values of the control variables
  • u(t 0 ) guess and u(t f ) guess are respectively
  • the guessed value of the input at the initial time and the end time are some initial conditions set.
  • C min and C max are the minimum and maximum values of the inequality path constraint parameters, respectively.
  • Figures 7a and 7b show the tension of two driving ropes that satisfy the path constraint of the inequality, with a minimum value of 0.05N and a maximum value of 0.3N.
  • the final joint angle is the same as the expected value, indicating that the influence of rope elasticity on the position accuracy is compensated, and the rope tension meets the constraint range, that is, the tension is greater than a certain value (minimum value), the driving rope will not slack, and the tension is less than a certain value. Value (maximum) to reduce the possibility of breakage of the drive rope.
  • the preferred embodiment of the present invention proposes a rope tension optimization method for a rope-driven linkage manipulator.
  • the tension optimization method comprehensively considers the dynamics of the manipulator, and the dynamic equation of the manipulator comprehensively considers the elasticity of the driving rope.
  • the end position error caused by the deformation of the driving rope is compensated, so that the rope tension is optimized: first, it is helpful to select the appropriate rope and guide the mechanism design; second, to reduce the rope during the movement through optimization The maximum tension can reduce the possibility of the rope breaking; third, make the rope always maintain a certain tension to prevent slack during the movement. Fourth, the end position error caused by the rope elasticity is compensated.

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Abstract

一种绳驱动联动式机械臂的动力学建模方法，包括：将机械臂等效为由刚体部分（10）、两根驱动绳索（20）和四根联动绳索（30）组成，刚体部分（10）包括三个旋转关节（11）和三根连杆（12），三根连杆（12）依次通过旋转关节（11）相互连接；两根驱动绳索（20）的一端分别固定在末端的连杆（12）上，另一端依次穿过各个连杆（12）后由驱动器控制；四根联动绳索（30）两两分别绕过相邻的旋转关节（11）以交叉地连接在连杆（12）上；建立机械臂的动力学方程。该方法实现了对机械臂的运动过程中绳索张力的优化，可用于绳驱动机械臂的轨迹规划、振动控制等方面。还涉及一种绳索张力优化方法。

Description

绳驱动联动式机械臂的动力学建模及其绳索张力优化方法 技术领域
本发明涉及绳驱动联动式机械臂领域，尤其涉及一种绳驱动联动式机械臂的动力学建模方法及其绳索张力优化方法。
背景技术
绳驱动联动式机械臂有两种控制模式：一种是张力控制模式，在绳索上安装力传感器作为反馈，直接控制绳索上的张力；另一种是绳长控制模式，将绳索看作轴向线性弹簧，通过调整绳索驱动规律间接调整绳索张力。无论哪种方式，都是绳索张力最终作用在连杆上，并驱动机器人运动。但是绳索张力受限于自身强度特性，主要是绳索材料及半径，以及绳索末端固定方式。此外绳索松弛也是常见的问题，绳索弹性降低了机械臂的末端精度。所有这些问题，在一定程度上限制了绳驱动机械臂的应用。
通过调整绳索长度，绳索张力会发生变化。但是在以往的研究中，绳索驱动长度规划往往从运动学层面出发，未考虑机械臂的动力学特性。而绳索弹性在机械臂的动力学中有着重要作用，一方面，通过将驱动器及减速器前置，绳驱动机械臂运动部分往往质量轻、惯量小，这种特性使得绳驱动机械臂具有高速响应的可能性；另一方面，绳索弹性可能使系统不稳定，并降低末端定位精度。因此，应考虑绳索弹性以充分挖掘绳驱动机器人的潜力。
而现有通常都是采用多项式插值或样条插值对绳索驱动规律进行规划，这存在以下几点问题：1、未考虑绳索弹性，未对绳索变形引起的末端位置误差进行补偿；2、难以确保运动过程中绳索始终处于张紧状态；3、仅仅从运动学角度进行规划，未考虑机械臂的动力学，可能使系统不稳定。
以上背景技术内容的公开仅用于辅助理解本发明的构思及技术方案，其并不必然属于本专利申请的现有技术，在没有明确的证据表明上述内容在本专利申请的申请日已经公开的情况下，上述背景技术不应当用于评价本申请的新颖性和创造性。
发明内容
为了解决上述技术问题，本发明提出一种绳驱动联动式机械臂的动力学建模方法及其绳索张力优化方法，实现了对机械臂的运动过程中绳索张力的优化，可用于绳驱动机械臂的轨迹规划、振动控制等方面。
为了达到上述目的，本发明采用以下技术方案：
本发明公开了一种绳驱动联动式机械臂的动力学建模方法，包括：
将所述绳驱动联动式机械臂等效为由刚体部分、两根驱动绳索和四根联动绳索组成，其中所述刚体部分包括三个旋转关节和三根连杆，三根所述连杆依次通过所述旋转关节相互连接；两根所述驱动绳索的一端分别固定在末端的所述连杆上，另一端依次穿过各个所述连杆后由驱动器控制；四根所述联动绳索两两分别绕过相邻的所述旋转关节以分别交叉地连接在所述连杆上；
建立所述绳驱动联动式机械臂的动力学方程为：
Figure PCTCN2020091146-appb-000001
其中，q为关节角，
Figure PCTCN2020091146-appb-000002
为关节角速度，
Figure PCTCN2020091146-appb-000003
为关节角加速度，M(q)为广义质量阵，
Figure PCTCN2020091146-appb-000004
为科氏力和向心力项，τ为关节力矩，τ c为联动绳索映射到旋转关节处的等效关节力矩。
优选地，其中关节力矩τ的表达式为：
τ＝H(q)T
其中，H(q)为驱动绳索的张力到关节力矩的映射关系，T为驱动绳索的张力；
驱动绳索的张力T的表达式为：
Figure PCTCN2020091146-appb-000005
其中，E a为驱动绳索的杨氏模量，A a为驱动绳索的截面积，l a(q)为当前时刻驱动绳索的长度，Δl a为驱动绳索的总变形量；
驱动绳索的总变形量Δl a的表达式为：
Δl a＝l a(q)-l a0-l am
其中，l a0为初始时刻驱动绳索的长度，l am为驱动器驱动的驱动绳索的长度。
优选地，其中联动绳索映射到旋转关节处的等效关节力矩τ c的表达式为：
τ c＝2k cr 2Δq
其中，k c为联动绳索的弹性系数，r为联动绳索的绕线半径，Δq为相邻的旋转关节的角度差；
联动绳索的弹性系数k c的表达式为：
Figure PCTCN2020091146-appb-000006
其中，E c为联动绳索的杨氏模量，A c为联动绳索的截面积，l c0为联动绳索的长度。
本发明还公开了一种对绳驱动联动式机械臂的驱动绳索进行张力优化的方法，包括：采用全局插值多项式估计状态变量与控制变量，对多项式求导得到状态变量的导数以将微分方程离散化，将最优控制问题转化为非线性约束问题，其中的多项式是指上述的动力学建模方法中建立的所述绳驱动联动式机械臂的动力学方程，得到的非线性约束问题包括动力学约束要求。
优选地，当所述绳驱动联动式机械臂为张力控制模式时，在第一非线性约束要求内寻求第一最小化目标函数的最优解以对所述绳驱动联动式机械臂的驱动绳索进行张力优化。
优选地，所述第一非线性约束要求包括动力学约束要求，且动力学约束要求的表达式为：
Figure PCTCN2020091146-appb-000007
其中，
Figure PCTCN2020091146-appb-000008
τ＝H(q)u，q为关节角，
Figure PCTCN2020091146-appb-000009
为关节角速度，
Figure PCTCN2020091146-appb-000010
为科氏力和向心力项，τ为关节力矩，τ c为联动绳索映射到关节处的等效关节力矩，M为广义质量阵，H(q)为驱动绳索的张力到关节力矩的映射关系，u为控制变量且u＝[T 1 T 2]，T 1、T 2分别表示两根驱动绳索的张力，x为状态变量且
Figure PCTCN2020091146-appb-000011
q＝[q 1 q 2 q 3]且q 1、q 2、q 3分别表示三个旋转关节的关节角，t表 示时间，t 0表示初始时刻，t f表示终止时刻；
所述第一最小化目标函数的表达式为：g＝max(u)。
优选地，所述第一非线性约束条件还包括边界条件约束要求和不等式路径约束要求；其中边界条件约束要求的表达式为：φ min≤φ(x(t 0),t 0,x(t f),t f)≤φ max，φ为边界条件约束参数；不等式路径约束要求的表达式为：C min≤C(x(t),u(t),t)≤C max，C为不等式路径约束参数。
优选地，当所述绳驱动联动式机械臂为绳长控制模式时，在满足第二非线性线性约束要求时即可对所述绳驱动联动式机械臂的驱动绳索进行张力优化。
优选地，所述第二非线性约束要求包括动力学约束要求，且动力学约束要求的表达式为：
Figure PCTCN2020091146-appb-000012
其中，
Figure PCTCN2020091146-appb-000013
τ＝H(q)T，
Figure PCTCN2020091146-appb-000014
q为关节角，
Figure PCTCN2020091146-appb-000015
为关节角速度，
Figure PCTCN2020091146-appb-000016
为科氏力和向心力项，τ为关节力矩，τ c为联动绳索映射到关节处的等效关节力矩，M为广义质量阵，H(q)为驱动绳索的张力到关节力矩的映射关系，u为控制变量且u＝[l 1 l 2]，l 1、l 2分别表示驱动器驱动的两根驱动绳索的长度，x为状态变量且
Figure PCTCN2020091146-appb-000017
q＝[q 1 q 2 q 3]且q 1、q 2、q 3分别表示三个旋转关节的关节角，t表示时间，t 0表示初始时刻，t f表示终止时刻，T表示驱动绳索的张力，E a为驱动绳索的杨氏模量，A a为驱动绳索的截面积，l a(q)为当前时刻驱动绳索的长度，l a0为初始时刻驱动绳索的长度。
优选地，所述第二非约束条件还包括边界条件约束要求和不等式路径约束要求；其中边界条件约束要求的表达式为：φ min≤φ(x(t 0),t 0,x(t f),t f)≤φ max，φ为边界条件约束参数；不等式路径约束要求的表达式为：T min≤T≤T max，T表示驱动绳索的张力。
与现有技术相比，本发明的有益效果在于：本发明提出的绳驱动联动式机械 臂的动力学建模方法综合考虑驱动绳索的弹性，因此对驱动绳索变形引起的末端位置误差进行了补偿。进一步地，本发明提出的对绳驱动联动式机械臂的驱动绳索进行张力优化的方法，综合考虑了机械臂的动力学，使得机械臂系统更加稳定，并确保了运动过程中绳索始终处于张紧状态；而且通过该张力优化方法对机械臂的驱动绳索进行了张力优化之后：第一，有助于选择合适的绳索及指导进行机构设计；第二，通过优化减小运动过程中绳索的最大张力，可以降低绳索破断的可能性；第三，使绳索始终保持一定的张力以防止运动过程中的松弛现象；第四，对由于绳索弹性引起的末端位置误差进行了补偿。
附图说明
图1a是本发明优选实施例的绳驱动联动式机械臂的伸直状态图；
图1b是图1a中的机械臂的等曲率弯曲状态图；
图2是图1a中的机械臂的联动绳索的示意图；
图3a和图3b分别为绳索张力控制模式下机械臂的两根驱动绳索的张力；
图4a和图4b分别为绳索张力控制模式下机械臂的关节角度和关节角速度；
图5a和图5b分别为绳长控制模式下机械臂的两根驱动绳索的驱动长度；
图6a和图6b分别为绳长控制模式下机械臂的关节角度和关节角速度；
图7a和图7b分别为绳长控制模式下机械臂的两根驱动绳索的张力。
具体实施方式
下面对照附图并结合优选的实施方式对本发明作进一步说明。
如图1a所示，是本发明优选实施例的绳驱动联动式机械臂的伸直状态，其中该机械臂由三部分组成，分别为刚体部分10、两根驱动绳索20和四根联动绳索30。其中刚体部分10由三个旋转关节11(包括第一旋转关节111、第二旋转关节112、第三旋转关节113)和三根连杆12(包括第一连杆121、第二连杆122、第三连杆123)组成，三根连杆12依次通过旋转关节11相互连接，连杆12在平面内运动。机械臂由两根独立驱动的驱动绳索20进行主动驱动，连杆12的两侧圆盘有绳孔，驱动绳索20穿过绳孔，末端通过绳头固定在末端的连杆12上，也即两根驱动绳索20的一端分别固定在末端的连杆12上，另一端依次穿过各个 连杆12后由电机驱动控制，通过电机拉紧驱动绳索20，驱动绳索20产生张力，在过孔处对连杆12产生作用力，驱动连杆12运动。四根联动绳索30两两分别绕过相邻的旋转关节11以交叉地连接在连杆12上(也即四根联动绳索30共分为两组两根联动绳索，其中一组的两根联动绳索301分别绕过第一旋转关节111和第二旋转关节112以交叉地设置在第一旋转关节111和第二旋转关节112之间的第一连杆121上，其中该两根联动绳索301的两端分别固定连接在基座40和第二连杆122上；另一组的两根联动绳索302分别绕过第二旋转关节112和第三旋转关节113以交叉地设置在第二旋转关节112和第三旋转关节113之间的第二连杆122上，其中该两根联动绳索302的两端分别固定连接在第一连杆121和第三连杆123上)，使得所有旋转关节11具有相同转角的机制，确保机械臂等曲率弯曲，如图1b所示。
下述建立该绳驱动联动式机械臂的动力学方程如下：
Figure PCTCN2020091146-appb-000018
其中，q为关节角，
Figure PCTCN2020091146-appb-000019
为关节角速度，
Figure PCTCN2020091146-appb-000020
为关节角加速度，M(q)为广义质量阵，
Figure PCTCN2020091146-appb-000021
为科氏力和向心力项，τ为关节力矩，τ c为联动绳索映射到旋转关节处的等效关节力矩。
如图2所示，是联动绳索30的示意图，联动绳索30的绕线半径为r，当相邻的旋转关节11有角度差Δq，则联动绳索30的变形量Δl c＝rΔq，设定联动绳索30的预紧力为T c0，则有：
Figure PCTCN2020091146-appb-000022
其中，T c1、T c2分别为两根联动绳索30的张力，τ c为联动绳索映射到旋转关节处的等效关节力矩，k c为联动绳索30的弹性系数且
Figure PCTCN2020091146-appb-000023
E c为联动绳索30的杨氏模量，A c为联动绳索30的截面积，l c0为联动绳索30的长度。
电机拉动驱动绳索20，驱动绳索20的张力随之变化，张力变化规律取决于 驱动绳索20的拉伸规律：
Figure PCTCN2020091146-appb-000024
其中，H(q)为驱动绳索20的张力到关节力矩的映射关系，T为驱动绳索20的张力，τ为关节力矩；E a为驱动绳索20的杨氏模量，A a为驱动绳索20的截面积，l a(q)为当前时刻驱动绳索20的长度，Δl a为驱动绳索20的总变形量；l a0为初始时刻驱动绳索20的长度，l am为电机驱动的驱动绳索20的长度，即电机拉动的驱动绳索20的长度。
下述结合绳驱动联动式机械臂的动力学方程对绳驱动联动式机械臂的驱动绳索的张力进行优化，其中驱动绳索的张力优化实质上为最优控制问题，具体实施中可以通过Matlab优化工具箱GPOPS进行优化。具体地：采用全局插值多项式估计状态变量与控制变量，对多项式求导得到状态变量的导数以将微分方程离散化，将最优控制问题转化为非线性约束问题，大大提高计算效率。其中的多项式是指上述建立的绳驱动联动式机械臂的动力学方程，得到的非线性约束问题包括动力学约束要求。
最小化目标函数为：
Figure PCTCN2020091146-appb-000025
其中，Φ为包括边界条件状态量的函数，g为包含整个过程状态量的积分项函数，t表示时间，t 0表示初始时刻，t f表示终止时刻，x为系统的状态变量，u为系统的控制变量。
需满足的动力学约束(动力学方程)、边界条件约束(初始时刻、终止时刻状态量范围)及不等式路径约束(运动过程中状态变量、控制变量的范围)分别为：
Figure PCTCN2020091146-appb-000026
其中，φ为边界条件约束参数，C为不等式路径约束参数。
(1)张力控制模式的绳驱动联动式机械臂
当绳驱动联动式机械臂为张力控制模式时，在第一非线性约束要求内寻求第一最小化目标函数的最优解，可以实现对绳驱动联动式机械臂的驱动绳索进行张力优化。
当绳驱动联动式机械臂为张力控制模式时，驱动绳索的张力作为系统的控制输入，如式(4)所示，目标函数包含两部分，Φ为包括边界条件状态量的函数，g为包含整个过程状态量的积分项函数；此时，目标函数只考虑第二项，并定义为驱动绳索的最大张力，即第一最小化目标函数为：
g＝max(u)   (6)
第一非约束要求包括动力学约束要求、边界条件约束要求和不等式路径约束要求，其中结合式(5)中的第一行的动力学约束条件表达式，张力控制模式下的动力学约束要求中需满足：
Figure PCTCN2020091146-appb-000027
其中的状态变量x和控制变量u分别为：
Figure PCTCN2020091146-appb-000028
其中，q为关节角，
Figure PCTCN2020091146-appb-000029
为关节角速度，
Figure PCTCN2020091146-appb-000030
为科氏力和向心力项，τ为关节力矩，τ c为联动绳索映射到关节处的等效关节力矩，M为广义质量阵，H(q)为驱动绳索的张力到关节力矩的映射关系，T 1、T 2分别表示两根驱动绳索的张力，q 1、q 2、q 3分别表示三个旋转关节的关节角。
边界条件约束要求和不等式路径约束要求分别为式(5)中的第二行和第三行的约束表达式。
(2)绳长控制模式的绳驱动联动式机械臂
当绳驱动联动式机械臂为绳长控制模式时，在满足第二非线性约束要求时即 可对绳驱动联动式机械臂的驱动绳索进行张力优化。
当绳驱动联动式机械臂为绳长控制模式时，绳索长度作为系统控制输入，绳索长度可以通过与电机编码器对应关系得到，这里通过不等式路径约束限制驱动绳索的张力范围。
第二非约束要求包括动力学约束要求、边界条件约束要求和不等式路径约束要求，其中结合式(5)中的第一行的动力学约束条件表达式，绳长控制模式下的动力学约束要求中需满足：
Figure PCTCN2020091146-appb-000031
其中的状态变量x和控制变量u分别为：
Figure PCTCN2020091146-appb-000032
其中，q为关节角，
Figure PCTCN2020091146-appb-000033
为关节角速度，
Figure PCTCN2020091146-appb-000034
为科氏力和向心力项，τ为关节力矩，τ c为联动绳索映射到关节处的等效关节力矩，M为广义质量阵，H(q)为驱动绳索的张力到关节力矩的映射关系，l 1、l 2分别表示电机驱动的两根驱动绳索的长度，q 1、q 2、q 3分别表示三个旋转关节的关节角，T表示驱动绳索的张力，E a为驱动绳索的杨氏模量，A a为驱动绳索的截面积，l a(q)为当前时刻驱动绳索的长度，l a0为初始时刻驱动绳索的长度。
边界条件约束要求为式(5)中的第二行的约束表达式。
不等式路径约束要求的表达式为：
T min≤T≤T max   (11)
其中，T表示驱动绳索的张力，T min是由驱动绳索不出现松弛现象的最小张力确定，T max是由绳索能承受的最大拉力确定。
从以上各式可以看出，上述两种模式的区别仅在于系统的输入不同，即公式 中的u(t)不同，张力控制模式中输入为绳索张力，绳长控制模式中输入为绳长变化；相应的动力学方程也有所不同，绳长控制模式相比张力控制模式增加了绳长变化到绳索张力的转化，具体如式(7)和式(9)。通过优化绳索张力，使驱动绳索张力保持在设定范围内，防止绳索松弛又减小绳索破断的可能性。
下述利用上述的张力优化方法进行仿真计算以对本发明优选实施例的对绳驱动联动式机械臂的驱动绳索进行张力优化的方法进行进一步说明。
如图1a和图1b所示，机械臂是由两根驱动绳索20驱动，三个旋转关节11通过四根联动绳索30进行联动，实现整个臂杆等曲率弯曲，连杆12在平面内运动。
连杆12的参数如下：质量m＝90g，质心坐标l c＝[0 0 33.7]mm，转动惯量I＝[65.73 62.05 19.08]g.mm 2。
联动绳索30的参数如下：绳索杨氏模量E c＝2.06e11Pa，绳索半径为R c＝0.3mm，绕线半径为r＝6.5mm。
驱动绳索20的参数如下：绳索杨氏模量E a＝2.1e9Pa，绳索的半径为R＝0.4mm。
(1)绳索张力控制模式：
边界及约束条件设置如下：
x(t 0)＝zeros(1,6)
x(t f)＝[-5*ones(1,3) zeros(1,3)]
x min＝[-6*ones(1,3) -10*ones(1,3)]
x max＝[6*ones(1,3) 10*ones(1,3)]
u min＝[0.05 0.05]
u max＝[0.22 0.22]
u(t 0) guess＝[0.05 0.05]
u(t f) guess＝[0.05 0.05]
其中，zeros(1,6)表示一行六列的0矩阵，ones(1,3)表示一行三列的1矩阵，x(t 0)、x(t f)分别为初始时刻、终止时刻的状态变量值；x min、x max分别为状态变 量的最小值、最大值，u min、u max分别为控制变量的最小值、最大值，u(t 0) guess、u(t f) guess分别为初始时刻、终止时刻输入量的猜测值，是设置的一些初始条件。
如图3a和图3b，是机械臂优化的控制变量输入，即两根驱动绳索的张力，整个过程中，两根绳索的张力的最大值为0.22N，最小值为0.05N；如图4a和图4b是机械臂优化的状态变量，图4a为得到的关节角度，终止时刻的关节角度都为-5°，图4b为得到的关节角速度，终止时刻的关节角速度为0，满足终止时刻约束条件。
(2)绳长控制模式：
边界及约束条件设置如下：
x(t 0)＝zeros(1,6)
x(t f)＝[-5*ones(1,3) zeros(1,3)]
x min＝[-6*ones(1,3) -12*ones(1,3)]
x max＝[6*ones(1,3) 12*ones(1,3)]
u min＝[0 -7.2]
u max＝[7.2 0]
u(t 0) guess＝[0 0]
u(t f) guess＝[5.4503 -5.5417]
C min＝[0.05 0.05]
C max＝[0.3 0.3]
其中，zeros(1,6)表示一行六列的0矩阵，ones(1,3)表示一行三列的1矩阵，x(t 0)、x(t f)分别为初始时刻、终止时刻的状态变量值；x min、x max分别为状态变量的最小值、最大值，u min、u max分别为控制变量的最小值、最大值，u(t 0) guess、u(t f) guess分别为初始时刻、终止时刻输入量的猜测值，是设置的一些初始条件，C min、C max分别是不等式路径约束参数的最小值、最大值。
如图5a和图5b，是机械臂优化的控制变量输入，即两根驱动绳索的长度，终止时刻驱动绳索的长度分别为[5.44585.5416]mm，可以看到与上式单纯运动学计算的u(t f) guess有细微差别，是因为考虑了驱动绳索的弹性；如图6a和图6b是机械臂优化的状态变量，图6a为得到的关节角度，图6b为得到的关节角速度， 终止时刻的关节速度、角速度满足约束条件。图7a和图7b分别为满足不等式路径约束的两根驱动绳索的张力，最小值为0.05N，最大值为0.3N。其中最终得到的关节角度与期望值相同，说明补偿了绳索弹性对位置精度的影响，而且绳索的张力满足约束范围，即张力大于一定值(最小值)，驱动绳索不会出现松弛现象，张力小于一定值(最大值)，减小驱动绳索的断裂可能性。
本发明优选实施例针对绳驱动联动式机械臂提出了一种绳索张力优化方法，该张力优化方法中综合考虑了机械臂的动力学，而机械臂的动力学方程中综合考虑驱动绳索的弹性，且对驱动绳索变形引起的末端位置误差进行了补偿，以使得通过对绳索张力进行优化：第一，有助于选择合适的绳索及指导进行机构设计；第二，通过优化减小运动过程中绳索的最大张力，可以降低绳索破断的可能性；第三，使绳索始终保持一定的张力以防止运动过程中的松弛现象。第四，对由于绳索弹性引起的末端位置误差进行了补偿。
以上内容是结合具体的优选实施方式对本发明所作的进一步详细说明，不能认定本发明的具体实施只局限于这些说明。对于本发明所属技术领域的技术人员来说，在不脱离本发明构思的前提下，还可以做出若干等同替代或明显变型，而且性能或用途相同，都应当视为属于本发明的保护范围。

Claims (10)

  1. 一种绳驱动联动式机械臂的动力学建模方法，其特征在于，包括：
    将所述绳驱动联动式机械臂等效为由刚体部分、两根驱动绳索和四根联动绳索组成，其中所述刚体部分包括三个旋转关节和三根连杆，三根所述连杆依次通过所述旋转关节相互连接；两根所述驱动绳索的一端分别固定在末端的所述连杆上，另一端依次穿过各个所述连杆后由驱动器控制；四根所述联动绳索两两分别绕过相邻的所述旋转关节以交叉地连接在所述连杆上；
    建立所述绳驱动联动式机械臂的动力学方程为：
    Figure PCTCN2020091146-appb-100001
    其中，q为关节角，
    Figure PCTCN2020091146-appb-100002
    为关节角速度，
    Figure PCTCN2020091146-appb-100003
    为关节角加速度，M(q)为广义质量阵，
    Figure PCTCN2020091146-appb-100004
    为科氏力和向心力项，τ为关节力矩，τ c为联动绳索映射到旋转关节处的等效关节力矩。
  2. 根据权利要求1所述的动力学建模方法，其特征在于，其中关节力矩τ的表达式为：
    τ＝H(q)T
    其中，H(q)为驱动绳索的张力到关节力矩的映射关系，T为驱动绳索的张力；
    驱动绳索的张力T的表达式为：
    Figure PCTCN2020091146-appb-100005
    其中，E a为驱动绳索的杨氏模量，A a为驱动绳索的截面积，l a(q)为当前时刻驱动绳索的长度，Δl a为驱动绳索的总变形量；
    驱动绳索的总变形量Δl a的表达式为：
    Δl a＝l a(q)-l a0-l am
    其中，l a0为初始时刻驱动绳索的长度，l am为驱动器驱动的驱动绳索的长度。
  3. 根据权利要求1所述的动力学建模方法，其特征在于，其中联动绳索映 射到旋转关节处的等效关节力矩τ c的表达式为：
    τ c＝2k cr 2Δq
    其中，k c为联动绳索的弹性系数，r为联动绳索的绕线半径，Δq为相邻的旋转关节的角度差；
    联动绳索的弹性系数k c的表达式为：
    Figure PCTCN2020091146-appb-100006
    其中，E c为联动绳索的杨氏模量，A c为联动绳索的截面积，l c0为联动绳索的长度。
  4. 一种对绳驱动联动式机械臂的驱动绳索进行张力优化的方法，其特征在于，包括：采用全局插值多项式估计状态变量与控制变量，对多项式求导得到状态变量的导数以将微分方程离散化，将最优控制问题转化为非线性约束问题，其中的多项式是指权利要求1至3任一项所述的动力学建模方法中建立的所述绳驱动联动式机械臂的动力学方程，得到的非线性约束问题包括动力学约束要求。
  5. 根据权利要求4所述的方法，其特征在于，当所述绳驱动联动式机械臂为张力控制模式时，在第一非线性约束要求内寻求第一最小化目标函数的最优解以对所述绳驱动联动式机械臂的驱动绳索进行张力优化。
  6. 根据权利要求5所述的方法，其特征在于，所述第一非线性约束要求包括动力学约束要求，且动力学约束要求的表达式为：
    Figure PCTCN2020091146-appb-100007
    其中，
    Figure PCTCN2020091146-appb-100008
    τ＝H(q)u，q为关节角，
    Figure PCTCN2020091146-appb-100009
    为关节角速度，
    Figure PCTCN2020091146-appb-100010
    为科氏力和向心力项，τ为关节力矩，τ c为联动绳索映射到关节处的等效关节力矩，M为广义质量阵，H(q)为驱动绳索的张力到关节力矩的映射关系，u为控制变量且u＝[T 1 T 2]，T 1、T 2分别表示两根驱动绳索的张力，x为状态变量且
    Figure PCTCN2020091146-appb-100011
    q＝[q 1 q 2 q 3]且q 1、q 2、q 3分别表示三个旋转关节的关节角，t表示时间，t 0表示初始时刻，t f表示终止时刻；
    所述第一最小化目标函数的表达式为：g＝max(u)。
  7. 根据权利要求6所述的方法，其特征在于，所述第一非线性约束条件还包括边界条件约束要求和不等式路径约束要求；其中边界条件约束要求的表达式为：φ min≤φ(x(t 0),t 0,x(t f),t f)≤φ max，φ为边界条件约束参数；不等式路径约束要求的表达式为：C min≤C(x(t),u(t),t)≤C max，C为不等式路径约束参数。
  8. 根据权利要求4所述的方法，其特征在于，当所述绳驱动联动式机械臂为绳长控制模式时，在满足第二非线性约束要求时即可对所述绳驱动联动式机械臂的驱动绳索进行张力优化。
  9. 根据权利要求8所述的方法，其特征在于，所述第二非线性约束要求包括动力学约束要求，且动力学约束要求的表达式为：
    Figure PCTCN2020091146-appb-100012
    其中，
    Figure PCTCN2020091146-appb-100013
    τ＝H(q)T，
    Figure PCTCN2020091146-appb-100014
    q为关节角，
    Figure PCTCN2020091146-appb-100015
    为关节角速度，
    Figure PCTCN2020091146-appb-100016
    为科氏力和向心力项，τ为关节力矩，τ c为联动绳索映射到关节处的等效关节力矩，M为广义质量阵，H(q)为驱动绳索的张力到关节力矩的映射关系，u为控制变量且u＝[l 1 l 2]，l 1、l 2分别表示驱动器驱动的两根驱动绳索的长度，x为状态变量且
    Figure PCTCN2020091146-appb-100017
    q＝[q 1 q 2 q 3]且q 1、q 2、q 3分别表示三个旋转关节的关节角，t表示时间，t 0表示初始时刻，t f表示终止时刻，T表示驱动绳索的张力，E a为驱动绳索的杨氏模量，A a为驱动绳索的截面积，l a(q)为当前时刻驱动绳索的长度，l a0为初始时刻驱动绳索的长度。
  10. 根据权利要求9所述的方法，其特征在于，所述第二非线性约束条件还包括边界条件约束要求和不等式路径约束要求；其中边界条件约束要求的表达式为：φ min≤φ(x(t 0),t 0,x(t f),t f)≤φ max，φ为边界条件约束参数；不等式路径约束要求的表达式为：T min≤T≤T max，T表示驱动绳索的张力。
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