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CN112757306A - Inverse solution multi-solution selection and time optimal trajectory planning algorithm for mechanical arm - Google Patents

Inverse solution multi-solution selection and time optimal trajectory planning algorithm for mechanical arm Download PDF

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CN112757306A
CN112757306A CN202110109515.5A CN202110109515A CN112757306A CN 112757306 A CN112757306 A CN 112757306A CN 202110109515 A CN202110109515 A CN 202110109515A CN 112757306 A CN112757306 A CN 112757306A
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CN112757306B (en
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陈光荣
郭盛
曲海波
陈亚琼
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Beijing Jiaotong University
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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
    • 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
    • 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
    • B25J9/1612—Programme controls characterised by the hand, wrist, grip control

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  • Robotics (AREA)
  • Mechanical Engineering (AREA)
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Abstract

本发明涉及足式机器人机械臂领域，特别涉及一种机械臂逆解多解选择和时间最优轨迹规划算法。该算法可以根据当前的机械臂位姿迅速从逆解多解中选择最合适的一组解，有效规避奇异位型，实现不同逆解之间的平滑切换，保证机械臂控制和动作的连续性和不振荡；并利用七次多项式和条件比例控制相结合的轨迹规划方法规划出时间最优的轨迹，保证关节角度、角速度、角加速度、角加加速度的平滑连续可导，同时满足多目标多约束条件，满足执行器规范，含关节角速度最大值约束或关节角加速度最大值约束等。该算法大大减小了多解选择和轨迹规划的计算量，可以实现实时高效的时间最优轨迹规划，并可推广到低算力控制器上应用。

Figure 202110109515

The invention relates to the field of a foot-type robot manipulator, in particular to an inverse solution multi-solution selection and time optimal trajectory planning algorithm of the manipulator. The algorithm can quickly select the most suitable set of solutions from the multiple inverse solutions according to the current position of the manipulator, effectively avoid singular positions, realize smooth switching between different inverse solutions, and ensure the continuity of control and action of the manipulator. and does not oscillate; and uses the trajectory planning method combined with the seventh-order polynomial and the conditional proportional control to plan the time-optimized trajectory to ensure the smooth and continuous derivation of the joint angle, angular velocity, angular acceleration, and angular jerk, while meeting the requirements of multiple objectives and multiple Constraints that meet the actuator specifications, including maximum joint angular velocity constraints or maximum joint angular acceleration constraints, etc. The algorithm greatly reduces the computational complexity of multi-solution selection and trajectory planning, can achieve real-time and efficient time-optimal trajectory planning, and can be extended to low-computing controllers.

Figure 202110109515

Description

Inverse solution multi-solution selection and time optimal trajectory planning algorithm for mechanical arm
Technical Field
The invention relates to the field of mechanical arms of foot type robots, in particular to an inverse solution multi-solution selection and time optimal trajectory planning algorithm for a mechanical arm.
Background
The mechanical arm is a complex system with high precision, multiple inputs and outputs, high nonlinearity and strong coupling. Because of its unique operational flexibility, it has been widely used in the fields of industrial assembly, safety and explosion protection. The mechanical arm is a complex system, and uncertainty such as parameter perturbation, external interference, unmodeled dynamic state and the like exists, so that uncertainty also exists in a modeling model of the mechanical arm. The mathematical model of the robot arm includes kinematics and dynamics, both of which include a positive solution and an inverse solution. Generally, multiple solutions exist in the inverse solution of the motion of the multi-degree-of-freedom mechanical arm, and for different tasks, the cartesian space or joint space trajectory of the mechanical arm needs to be planned and used as a target given signal of the mechanical arm for control. How to select the most appropriate solution group from the inverse solution multi-solutions and plan the optimal time trajectory in a concise and efficient manner is very much concerned in both theoretical research and practical application. According to the inverse solution multi-solution selection and time optimal trajectory planning algorithm for the mechanical arm, the most appropriate one group of solutions can be rapidly selected from the inverse solution multi-solutions according to the current pose of the mechanical arm, the singular position type is effectively avoided, smooth switching among different inverse solutions is realized, and the continuity and non-oscillation of mechanical arm control and action are ensured; and a trajectory planning method combining the seventh polynomial and the conditional proportional control is utilized to plan a trajectory with optimal time, so that smooth and continuous guidance of joint angles, angular velocities, angular accelerations and angular jerks is ensured, multi-target and multi-constraint conditions are met, the actuator specification is met, and the maximum constraint of the joint angular velocities or the maximum constraint of the joint angular accelerations is included. The algorithm greatly reduces the calculation amount of multi-solution selection and trajectory planning, can realize real-time and efficient time optimal trajectory planning, and can be popularized and applied to low-calculation force controllers.
Disclosure of Invention
The invention aims to provide an inverse solution multi-solution selection and time optimal trajectory planning algorithm for a mechanical arm, which is simple and efficient in algorithm architecture and has the capability of quickly selecting the most appropriate group of solutions from the inverse solution multi-solutions and planning the time optimal trajectory.
The technical scheme adopted by the invention for solving the technical problems is as follows:
the mechanical arm inverse solution multi-solution selection and time optimal trajectory planning algorithm is characterized in that: on the basis of a kinematic forward solution and a kinematic inverse solution, the overall algorithm framework of the method is composed of: the method comprises an inverse solution multi-solution selection algorithm and a time optimal trajectory planning algorithm. The general idea of solving the multi-solution selection algorithm is as follows: and performing forward-inverse solution or inverse-forward solution conversion according to the current pose of the mechanical arm, and taking the inverse solution with the minimum norm of the difference between the conversion result and the current pose of the mechanical arm as the most appropriate solution. The overall idea of the time optimal trajectory planning algorithm is as follows: and (3) carrying out track planning by using a seventh polynomial track planning method, calculating values corresponding to all constraint conditions under the track, verifying whether the values are equal to constraint condition thresholds or not, if not, taking the difference value between the values and the corresponding constraint condition thresholds to carry out proportional control and negative feedback to the original track planning time so as to reduce the difference value, and simultaneously, distinguishing and determining whether the negative feedback is carried out on different constraint conditions or not by using activation conditions.
Taking the point-to-point trajectory planning of a five-degree-of-freedom mechanical arm in a Cartesian space as an example, the algorithm comprises the following four steps:
the first step is as follows: a kinematic positive solution.
Firstly, establishing a mechanical arm base coordinate system {0}, each joint coordinate system {1,2.3.4} and a terminal coordinate system {5}, and establishing a mechanical arm D-H parameter table according to mechanical arm structure parameters;
TABLE 1.1D-H PARAMETER TABLE FOR MECHANICAL ARM
Figure BDA0002914706100000021
Secondly, obtaining a homogeneous transformation matrix among the coordinates according to the established coordinate system and the D-H parameter table of the mechanical arm; from the coordinate system oi-1-xi-1yi-1zi-1To the coordinate system oi-xiyiziIs transformed into
Figure BDA0002914706100000031
Thirdly, according to the homogeneous transformation matrix between the base coordinate system and the tail end coordinate system of the mechanical arm
Figure BDA0002914706100000032
The position of the robot arm tip relative to the base coordinate system and the pose of the robot arm tip relative to the base coordinate system are obtained, i.e., the kinematic positive solution p of the robot arm is Forward _ kinematics (θ), where p is [ x y z α β γ ═ y]To know the pose of the end of the arm, θ ═ θ1 θ2 θ3 θ4 θ5]The calculated joint angles of the robot.
The second step is that: inverse kinematics solution.
(i) can be obtained from the formula (2)
Figure BDA0002914706100000033
The first column and the fourth column of equation (2) are made equal, so that six equations can be obtained;
based on the six equations obtained above, the equation can be solved in turn to obtain theta1Two solutions of, each theta1Can determine theta5Two solutions of (each group of (theta)1,θ5) Can determine theta3Two solutions of (a) and then finding theta1A solution of (a) and theta4A solution of (a); summarizing, a total of eight solutions, i.e., Inverse kinematics solution θ of the mechanical arm — Inverse of Inverse kinematics (i, p), (i ∈ [1,8]), can be obtained])。
The third step: trajectory planning based on a seventh order polynomial.
In order to ensure that the acceleration is continuous and avoid the shock caused by sudden acceleration when the mechanical arm is started and stopped, a seventh-order polynomial can be adopted for optimization, and the optimization aim is to ensure that the acceleration is continuously derivable, namely the acceleration of a planning starting point and a terminal point of a constraint track is large. The specific mechanical arm tail end track planning formula is
Figure BDA0002914706100000041
Secondly, constraint conditions are given, namely the positions, the speeds, the angular speeds and the acceleration of the starting point and the end point are respectively as follows:
Figure BDA0002914706100000042
substituting formula (5) for formula (4) to obtain the coefficients of a seventh-order polynomial
Figure BDA0002914706100000043
Namely, the seven-degree polynomial track planning which aims at acceleration continuous guidance is completed.
The fourth step: and optimizing the track.
The inverse solution multi-solution selection algorithm: first, a set of solutions i ═ 1 is sequentially selected from eight sets of solutions of the inverse arm solution, (i ∈ [1,8], and]) (ii) a Then, reading the current joint angle theta of the mechanical arm ═ theta1 θ2 θ3 θ4 θ5]Carrying out forward solution according to the current pose of the mechanical arm to obtain the current end pose, and carrying out inverse solution according to the current end pose to obtain each joint angle theta under the i-th group of inverse solutionsi＝[θi1 θi2 θi3 θi4 θi5]Finally compare | θi-θ|≤εθIn which epsilonθFor self-defining an angle threshold value, the method is used for distinguishing the most appropriate solution from eight sets of inverse solutions, if thetai-θ|≤εθIf the solution is satisfied, the solution is the optimal solution; if not, another group of inverse solutions is taken for recalculation and comparison;
time optimal trajectory optimization algorithm based on condition proportional control
The algorithm can be divided into two cases:
case 1: the angular velocity of the planned trajectory obtained in the third step
Figure BDA0002914706100000051
And angular acceleration
Figure BDA0002914706100000052
Although all meet the actuator specification, i.e. the maximum angular velocity at which the joint can execute
Figure BDA0002914706100000053
And angular acceleration
Figure BDA0002914706100000054
But with a planning time T equal to Tf-t0Longer, not optimal, where t0,tfThe times of passing the starting point and the ending point, respectively. At this point, the planning time needs to be reduced to achieve time optimization while meeting actuator specifications.
Case 2: the angular velocity of the planned trajectory obtained in the third step
Figure BDA0002914706100000055
And angular acceleration
Figure BDA0002914706100000056
Albeit greater than the actuator specification, i.e. the maximum angular velocity at which the joint can execute
Figure BDA0002914706100000057
And angular acceleration
Figure BDA0002914706100000058
The simplest solution at this time is to increase the planning time to achieve time optimization while meeting the actuator specifications.
According to the two situations, the time controller based on the condition proportional control can be designed without iteration, is simple and easy to realize and combines the proportional-integral-derivative (PID) control principle:
Figure BDA0002914706100000059
wherein
Figure BDA00029147061000000510
The adjusting parameters are respectively the error adjusting parameters of the speed and the acceleration in the time optimal controller, and are similar to the proportional parameters in the PID control principle. Whether the proportional control in equation (7) is activated depends on whether its corresponding actuator specification is satisfied, i.e., the origin of the conditional proportional control. In particular, it is possible to use, for example,
Figure BDA00029147061000000511
and
Figure BDA00029147061000000512
may be defined individually by joints or axes. Under ideal conditions, the planning time T is T ═ Tf-t0When the optimal angular velocity and angular acceleration of the planned trajectory are equal to the maximum angular velocity and angular acceleration allowed by the actuator specifications, i.e. the angular velocity and angular acceleration of the planned trajectory are equal to the maximum angular velocity and angular acceleration allowed by the actuator specifications
Figure BDA0002914706100000061
In addition, according to the requirements of multiple constraints and multiple targets, the maximum jerk term and other kinematic and dynamic constraint terms are required to be added into the right formula (7). In order to improve the transient performance of the time-optimal controller, integral control and derivative control are also required to be added with reference to PID control.
The algorithm must first discuss case 1 and discuss case 2 to ensure that constraints such as actuator specifications are fully satisfied. Firstly, initializing planning parameters: planning starting point p0Planning an end point pfPlanning exercise time T ═ Tf-t0(ii) a Secondly, interpolating and planning the track based on a seventh-order polynomial:
Figure BDA0002914706100000062
the resulting inverse solution is again used to find each joint angle:
Figure BDA0002914706100000063
finally, whether situation 1 is satisfied is determined:
if so, the planning time T-T is reduced by equation (7)f-t0Replanning the track; if not, case 2 is entered. In case 2, as in case 1, first based onAnd (3) a seventh-order polynomial interpolation planning track:
Figure BDA0002914706100000064
the resulting inverse solution is again used to find each joint angle:
Figure BDA0002914706100000065
finally, whether situation 2 is satisfied is determined: if so, increasing the programming time T-T by equation (7)f-t0Replanning the track; if not, case 2 exits. The two case discussion ends.
Output (c)
Figure BDA0002914706100000066
The algorithm ends as a given of the joint control.
In the algorithm, the order of polynomial interpolation track planning is determined by specific track planning requirements, and is not limited to seven times; in joint space, in continuous trajectory planning, the algorithm can obtain the same effect; the algorithm is also suitable for other multi-degree-of-freedom mechanical arms or series robots with multiple inverse solutions.
Compared with the prior art, the invention has the following beneficial effects: the inverse solution multi-solution selection and time optimal trajectory planning algorithm for the mechanical arm, provided by the invention, does not adopt an iteration principle, is simple and efficient in algorithm architecture, has the capability of quickly selecting the most appropriate one group of solutions from the inverse solution multi-solutions and planning the time optimal trajectory, can quickly select the most appropriate one group of solutions from the inverse solution multi-solutions according to the current pose of the mechanical arm, effectively avoids a singular position type, realizes smooth switching among different inverse solutions, and ensures the continuity and non-oscillation of control and action of the mechanical arm; and a trajectory planning method combining the seventh polynomial and the conditional proportional control is utilized to plan a trajectory with optimal time, so that smooth and continuous guidance of joint angles, angular velocities, angular accelerations and angular jerks is ensured, multi-target and multi-constraint conditions are met, and the actuator specifications, such as joint angular velocity maximum value constraint or joint angular acceleration maximum value constraint, are met. The algorithm greatly reduces the calculation amount of multi-solution selection and trajectory planning, can realize real-time and efficient time optimal trajectory planning, and can be popularized and applied to low-calculation force controllers.
Drawings
FIG. 1 is a schematic diagram of a five degree-of-freedom robotic arm;
FIG. 2 is a schematic diagram of structural parameters and a coordinate system of a five-degree-of-freedom mechanical arm;
FIG. 3 is a flow chart of the algorithm;
FIG. 4 illustrates the joint angles of the robot arm after the track optimization;
FIG. 5 shows the angular velocity of each joint of the mechanical arm after the track optimization;
FIG. 6 shows angular accelerations of joints of the mechanical arm after track optimization;
FIG. 7 illustrates the jerk of each joint angle of the mechanical arm after the trajectory optimization;
FIG. 8 track optimized end of arm position;
FIG. 9 trajectory optimized robotic arm tip pose;
FIG. 10 is a schematic diagram of a time optimization process for an initial planning time 10 s;
FIG. 11 is a schematic diagram of a time optimization process for the initial planning time 20 s;
Detailed Description
The invention is further explained with reference to the drawings.
The invention aims to provide an inverse solution multi-solution selection and time optimal trajectory planning algorithm for a mechanical arm, which is simple and efficient in algorithm architecture and has the capability of quickly selecting the most appropriate group of solutions from the inverse solution multi-solutions and planning the time optimal trajectory.
The technical scheme adopted by the invention for solving the technical problems is as follows:
the mechanical arm inverse solution multi-solution selection and time optimal trajectory planning algorithm is characterized in that: on the basis of a kinematic forward solution and a kinematic inverse solution, the overall algorithm framework of the method is composed of: the method comprises an inverse solution multi-solution selection algorithm and a time optimal trajectory planning algorithm. The general idea of solving the multi-solution selection algorithm is as follows: and performing forward-inverse solution or inverse-forward solution conversion according to the current pose of the mechanical arm, and taking the inverse solution with the minimum norm of the difference between the conversion result and the current pose of the mechanical arm as the most appropriate solution. The overall idea of the time optimal trajectory planning algorithm is as follows: and (3) carrying out track planning by using a seventh polynomial track planning method, calculating values corresponding to all constraint conditions under the track, verifying whether the values are equal to constraint condition thresholds or not, if not, taking the difference value between the values and the corresponding constraint condition thresholds to carry out proportional control and negative feedback to the original track planning time so as to reduce the difference value, and simultaneously, distinguishing and determining whether the negative feedback is carried out on different constraint conditions or not by using activation conditions.
As shown in fig. 1, taking a five-degree-of-freedom mechanical arm to perform point-to-point trajectory planning in cartesian space as an example, the algorithm includes the following four steps:
the first step is as follows: a kinematic positive solution.
Firstly, as shown in FIG. 2, establishing a mechanical arm basic coordinate system {0}, each joint coordinate system {1,2.3.4} and a terminal coordinate system {5}, and establishing a mechanical arm D-H parameter table according to mechanical arm structure parameters;
TABLE 1.1D-H PARAMETER TABLE FOR MECHANICAL ARM
Figure BDA0002914706100000081
Secondly, obtaining a homogeneous transformation matrix among the coordinates according to the established coordinate system and the D-H parameter table of the mechanical arm; from the coordinate system oi-1-xi-1yi-1zi-1To the coordinate system oi-xiyiziIs transformed into
Figure BDA0002914706100000082
Thirdly, according to the homogeneous transformation matrix between the base coordinate system and the tail end coordinate system of the mechanical arm
Figure BDA0002914706100000083
Obtaining a coordinate system of the end of the robot arm relative to the baseThe position of the robot arm tip and the attitude of the robot arm tip relative to the base coordinate system, i.e., the kinematic positive solution of the robot arm, p, Forward _ kinematics (θ), where p is [ x y z α β γ ═ y]To know the pose of the end of the arm, θ ═ θ1 θ2 θ3 θ4 θ5]The calculated joint angles of the robot.
The second step is that: inverse kinematics solution.
(i) can be obtained from the formula (2)
Figure BDA0002914706100000091
The first column and the fourth column of equation (2) are made equal, so that six equations can be obtained;
based on the six equations obtained above, the equation can be solved in turn to obtain theta1Two solutions of, each theta1Can determine theta5Two solutions of (each group of (theta)1,θ5) Can determine theta3Two solutions of (a) and then finding theta1A solution of (a) and theta4A solution of (a); summarizing, a total of eight solutions, i.e., Inverse kinematics solution θ of the mechanical arm — Inverse of Inverse kinematics (i, p), (i ∈ [1,8]), can be obtained])。
The third step: trajectory planning based on a seventh order polynomial.
In order to ensure that the acceleration is continuous and avoid the shock caused by sudden acceleration when the mechanical arm is started and stopped, a seventh-order polynomial can be adopted for optimization, and the optimization aim is to ensure that the acceleration is continuously derivable, namely the acceleration of a planning starting point and a terminal point of a constraint track is large. The specific mechanical arm tail end track planning formula is
Figure BDA0002914706100000092
Secondly, constraint conditions are given, namely the positions, the speeds, the angular speeds and the acceleration of the starting point and the end point are respectively as follows:
Figure BDA0002914706100000101
substituting formula (5) for formula (4) to obtain the coefficients of a seventh-order polynomial
Figure BDA0002914706100000102
Namely, the seven-degree polynomial track planning which aims at acceleration continuous guidance is completed.
The fourth step: and optimizing the track.
The inverse solution multi-solution selection algorithm: first, a set of solutions i ═ 1 is sequentially selected from eight sets of solutions of the inverse arm solution, (i ∈ [1,8], and]) (ii) a Then, reading the current joint angle theta of the mechanical arm ═ theta1 θ2 θ3 θ4 θ5]Carrying out forward solution according to the current pose of the mechanical arm to obtain the current end pose, and carrying out inverse solution according to the current end pose to obtain each joint angle theta under the i-th group of inverse solutionsi＝[θi1 θi2 θi3 θi4 θi5]Finally compare | θi-θ|≤εθIn which epsilonθFor self-defining an angle threshold value, the method is used for distinguishing the most appropriate solution from eight sets of inverse solutions, if thetai-θ|≤εθIf the solution is satisfied, the solution is the optimal solution; if not, another group of inverse solutions is taken for recalculation and comparison;
time optimal trajectory optimization algorithm based on condition proportional control
The algorithm can be divided into two cases:
case 1: the angular velocity of the planned trajectory obtained in the third step
Figure BDA0002914706100000111
And angular acceleration
Figure BDA0002914706100000112
Although all meet the actuator specification, i.e. the maximum angular velocity at which the joint can execute
Figure BDA0002914706100000113
Angle of harmonyAcceleration of a vehicle
Figure BDA0002914706100000114
But with a planning time T equal to Tf-t0Longer, not optimal, where t0,tfThe times of passing the starting point and the ending point, respectively. At this point, the planning time needs to be reduced to achieve time optimization while meeting actuator specifications.
Case 2: the angular velocity of the planned trajectory obtained in the third step
Figure BDA0002914706100000115
And angular acceleration
Figure BDA0002914706100000116
Albeit greater than the actuator specification, i.e. the maximum angular velocity at which the joint can execute
Figure BDA0002914706100000117
And angular acceleration
Figure BDA0002914706100000118
The simplest solution at this time is to increase the planning time to achieve time optimization while meeting the actuator specifications.
According to the two situations, the time controller based on the condition proportional control can be designed without iteration, is simple and easy to realize and combines the proportional-integral-derivative (PID) control principle:
Figure BDA0002914706100000119
wherein
Figure BDA00029147061000001110
The adjusting parameters are respectively the error adjusting parameters of the speed and the acceleration in the time optimal controller, and are similar to the proportional parameters in the PID control principle. Whether the proportional control in equation (7) is activated depends on whether its corresponding actuator specification is satisfied, i.e., the origin of the conditional proportional control. In particular, it is possible to use, for example,
Figure BDA00029147061000001111
and
Figure BDA00029147061000001112
may be defined individually by joints or axes. Under ideal conditions, the planning time T is T ═ Tf-t0When the optimal angular velocity and angular acceleration of the planned trajectory are equal to the maximum angular velocity and angular acceleration allowed by the actuator specifications, i.e. the angular velocity and angular acceleration of the planned trajectory are equal to the maximum angular velocity and angular acceleration allowed by the actuator specifications
Figure BDA00029147061000001113
In addition, according to the requirements of multiple constraints and multiple targets, the maximum jerk term and other kinematic and dynamic constraint terms are required to be added into the right formula (7). In order to improve the transient performance of the time-optimal controller, integral control and derivative control are also required to be added with reference to PID control.
The algorithm must first discuss case 1 and discuss case 2 to ensure that constraints such as actuator specifications are fully satisfied. Firstly, initializing planning parameters: planning starting point p0Planning an end point pfPlanning exercise time T ═ Tf-t0(ii) a Secondly, interpolating and planning the track based on a seventh-order polynomial:
Figure BDA0002914706100000121
the resulting inverse solution is again used to find each joint angle:
Figure BDA0002914706100000122
finally, whether situation 1 is satisfied is determined:
if so, the planning time T-T is reduced by equation (7)f-t0Replanning the track; if not, case 2 is entered. In case 2, as in case 1, the trajectory is first interpolated based on a seventh order polynomial:
Figure BDA0002914706100000123
the resulting inverse solution is again used to find each joint angle:
Figure BDA0002914706100000124
finally, whether situation 2 is satisfied is determined: if so, increasing the programming time T-T by equation (7)f-t0Replanning the track; if not, case 2 exits. The two case discussion ends.
Output (c)
Figure BDA0002914706100000125
The algorithm ends as a given of the joint control.
The algorithm flow chart is shown in fig. 3. Fig. 4-9 are the results of the trajectory planning optimization from point to point in cartesian space, which are: each joint angle, each joint angular velocity, each joint angular acceleration, each joint angular jerk, the tip position, and the tip attitude. Fig. 10-11 are time optimization processes with initial planning times of 10s and 20s, respectively.
In the algorithm, the order of polynomial interpolation track planning is determined by specific track planning requirements, and is not limited to seven times; in joint space, in continuous trajectory planning, the algorithm can obtain the same effect; the algorithm is also suitable for other multi-degree-of-freedom mechanical arms or series robots with multiple inverse solutions.
Compared with the prior art, the invention has the following beneficial effects: the inverse solution multi-solution selection and time optimal trajectory planning algorithm for the mechanical arm, provided by the invention, does not adopt an iteration principle, is simple and efficient in algorithm architecture, has the capability of quickly selecting the most appropriate one group of solutions from the inverse solution multi-solutions and planning the time optimal trajectory, can quickly select the most appropriate one group of solutions from the inverse solution multi-solutions according to the current pose of the mechanical arm, effectively avoids a singular position type, realizes smooth switching among different inverse solutions, and ensures the continuity and non-oscillation of control and action of the mechanical arm; and a trajectory planning method combining the seventh polynomial and the conditional proportional control is utilized to plan a trajectory with optimal time, so that smooth and continuous guidance of joint angles, angular velocities, angular accelerations and angular jerks is ensured, multi-target and multi-constraint conditions are met, and the actuator specifications, such as joint angular velocity maximum value constraint or joint angular acceleration maximum value constraint, are met. The algorithm greatly reduces the calculation amount of multi-solution selection and trajectory planning, can realize real-time and efficient time optimal trajectory planning, and can be popularized and applied to low-calculation force controllers.

Claims (3)

1.一种机械臂逆解多解选择和时间最优轨迹规划算法，其特征在于：在运动学正解和逆解的基础上，它的总体算法架构由：逆解多解选择算法和时间最优轨迹规划算法，两大部分组成；其中，解多解选择算法的总体思路为：根据当前的机械臂位姿进行正-逆解或者逆-正解转换，取转换结果与当前机械臂位姿的差的范数最小的逆解作为最合适的一组解；时间最优轨迹规划算法的总体思路为：利用七次多项式轨迹规划方法进行轨迹规划，计算该轨迹下的各约束条件对应的值，并验证该值是否等于约束条件阈值，若不等于，则取该值与对应约束条件阈值的差值进行比例控制负反馈到原来的轨迹规划时间上，以缩小该差值，同时利用激活条件来区分和决定不同约束条件是否进行负反馈；1. A manipulator inverse solution multi-solution selection and time optimal trajectory planning algorithm is characterized in that: on the basis of the kinematics positive solution and inverse solution, its overall algorithm architecture is composed of: inverse solution multi-solution selection algorithm and time maximum. The optimal trajectory planning algorithm consists of two parts; among them, the general idea of the multi-solution selection algorithm is: perform forward-inverse solution or inverse-positive solution conversion according to the current robot arm pose, and take the difference between the conversion result and the current robot arm pose. The inverse solution with the smallest norm of the difference is regarded as the most suitable set of solutions; the general idea of the time-optimal trajectory planning algorithm is: use the seven-order polynomial trajectory planning method to plan the trajectory, and calculate the value corresponding to each constraint condition under the trajectory, And verify whether the value is equal to the constraint condition threshold, if not, take the difference between the value and the corresponding constraint condition threshold for proportional control and negative feedback to the original trajectory planning time to reduce the difference, and use the activation condition to Distinguish and decide whether different constraints are negative feedback; 以一个五自由度机械臂在笛卡尔空间进行点到点的轨迹规划为例，算法的步骤分为以下四步：Taking the point-to-point trajectory planning of a five-degree-of-freedom robotic arm in Cartesian space as an example, the steps of the algorithm are divided into the following four steps: 第一步：运动学正解；The first step: correct kinematics solution; ①建立机械臂结基座标系{0}、各关节坐标系{1,2.3.4}和末端坐标系{5}，根据机械臂结构参数建立机械臂D-H参数表；①Establish the base frame of the manipulator {0}, the coordinate system of each joint {1, 2.3.4} and the end coordinate system {5}, and establish the D-H parameter table of the manipulator according to the structural parameters of the manipulator; 表1.1机械臂D-H参数表Table 1.1 D-H parameter table of robot arm
Figure FDA0002914706090000011
Figure FDA0002914706090000011
 ②根据建立的坐标系和机械臂D-H参数表，得出各坐标之间的齐次变换矩阵；从坐标系oi-1-xi-1yi-1zi-1到坐标系oi-xiyizi的变换矩阵为②According to the established coordinate system and the DH parameter table of the manipulator, the homogeneous transformation matrix between the coordinates is obtained; from the coordinate system o i-1 -x i-1 y i-1 z i-1 to the coordinate system o i The transformation matrix of -x i y i z i is
Figure FDA0002914706090000021
Figure FDA0002914706090000021
 ③根据机械臂结基座标系和末端坐标系之间的齐次变换矩阵③According to the homogeneous transformation matrix between the base coordinate system and the end coordinate system of the robot arm
Figure FDA0002914706090000022
Figure FDA0002914706090000022
 得出机械臂末端相对于基座标系的位置和机械臂末端相对于基座标系的姿态，即机械臂的运动学正解p＝Forward_kinematics(θ)，其中，p＝[x y z α β γ]为已知得机械臂末端位姿，θ＝[θ1 θ2 θ3 θ4 θ5]为所求的机器人各关节角度；Obtain the position of the end of the manipulator relative to the base frame and the attitude of the end of the manipulator relative to the base frame, that is, the positive kinematics solution of the manipulator p=Forward_kinematics(θ), where, p=[xyz α β γ] In order to obtain the known pose of the end of the manipulator, θ=[θ 1 θ 2 θ 3 θ 4 θ 5 ] is the desired angle of each joint of the robot; 第二步：运动学逆解；The second step: inverse kinematics solution; ①由(2)式可得①From the formula (2), we can get
Figure FDA0002914706090000023
Figure FDA0002914706090000023
 令式(2)的第一列和第四列相等，可得六个等式；Make the first and fourth columns of Equation (2) equal, six equations can be obtained; ②根据以上得到的六个等式，可依次求解得θ1的两个解，每个θ1可求得θ5的两个解，每组(θ1,θ5)可求得θ3的两个解，再求得θ1的一个解和θ4的一个解；②According to the six equations obtained above, two solutions of θ 1 can be solved in turn, two solutions of θ 5 can be obtained for each θ 1 , and two solutions of θ 3 can be obtained for each group (θ 1 , θ 5 ) Two solutions, and then a solution for θ 1 and a solution for θ 4 are obtained; ③综上，总共可得到八个解，即机械臂的运动学逆解θ＝Inverse_kinematics(i,p),(i∈[1,8])；③To sum up, a total of eight solutions can be obtained, that is, the inverse kinematics solution of the manipulator θ=Inverse_kinematics(i,p),(i∈[1,8]); 第三步：基于七次多项式的轨迹规划；The third step: trajectory planning based on the seventh degree polynomial; ①为了使得加速度也连续，避免机械臂启动和停止时加速度突变产生震动，可采用七次多项式进行优化，优化目标为保证加速度连续可导，即约束轨迹规划起点和终点的加加速度大小；具体机械臂末端轨迹规划公式为①In order to make the acceleration continuous and avoid vibration caused by the sudden acceleration of the manipulator when the manipulator starts and stops, the seventh-order polynomial can be used for optimization. The trajectory planning formula of the arm end is:
Figure FDA0002914706090000031
Figure FDA0002914706090000031
 ②给定约束条件，即起点和终点的位置，速度，角速度，加加速度分别为：② Given constraints, that is, the position, velocity, angular velocity, and jerk of the starting point and end point are:
Figure FDA0002914706090000032
Figure FDA0002914706090000032
 ③将式(5)代入式(4)可求得七次多项式系数分别为③ Substituting Equation (5) into Equation (4), the coefficients of the seventh-order polynomial can be obtained as
Figure FDA0002914706090000033
Figure FDA0002914706090000033
 即完成了以加速度连续可导为目标的七次多项式轨迹规划；That is, the seventh-order polynomial trajectory planning with the acceleration continuously derivable as the goal is completed; 第四步：轨迹优化；The fourth step: trajectory optimization; ①逆解多解选择算法：首先，在机械臂逆解的八组解中依次选择一组解i＝1,(i∈[1,8])；然后，读取机械臂当前各关节角θ＝[θ1 θ2 θ3 θ4 θ5]，根据当前的机械臂位姿进行正解得到当前的末端位姿，再根据得到当前的末端位姿进行逆解得到第i组逆解下的各关节角度θi＝[θi1 θi2 θi3 θi4 θi5]，最后比较|θi-θ|≤εθ，其中εθ为自定义角度阈值，用于从八组逆解中区分出最合适的解，若|θi-θ|≤εθ满足，则该解为最优解；若不满足，则取另一组逆解重新计算并比较；①Inverse solution multi-solution selection algorithm: First, select a group of solutions i=1, (i∈[1,8]) in turn from the eight groups of solutions of the inverse solution of the manipulator; then, read the current joint angle θ of the manipulator =[θ 1 θ 2 θ 3 θ 4 θ 5 ], perform a positive solution according to the current robot arm pose to obtain the current end pose, and then perform an inverse solution according to the obtained current end pose to obtain each of the i-th inverse solutions. Joint angle θ i = [θ i1 θ i2 θ i3 θ i4 θ i5 ], and finally compare |θ i -θ|≤ε θ , where ε θ is a custom angle threshold, which is used to distinguish the most Appropriate solution, if |θ i -θ|≤ε θ is satisfied, then the solution is the optimal solution; if not, another set of inverse solutions are recalculated and compared; ②基于条件比例控制的时间最优轨迹优化算法② Time optimal trajectory optimization algorithm based on conditional proportional control 该算法可分为两种情形：The algorithm can be divided into two cases: 情形1：第三步得到的规划轨迹的角速度
Figure FDA0002914706090000041
和角加速度
Figure FDA0002914706090000042
虽然都满足执行器规范，即关节可执行的最大角速度
Figure FDA0002914706090000043
和角加速度
Figure FDA0002914706090000044
但其规划时间T＝tf-t0比较长，并不是最优的，其中t0,tf分别为经过起始点和终止点的时间；此时，需要缩小规划时间以达到时间最优同时又满足执行器规范；Case 1: The angular velocity of the planned trajectory obtained in the third step
Figure FDA0002914706090000041
and angular acceleration
Figure FDA0002914706090000042
Although both meet the actuator specification, that is, the maximum angular velocity that the joint can perform
Figure FDA0002914706090000043
and angular acceleration
Figure FDA0002914706090000044
However, its planning time T=t f -t 0 is relatively long, which is not optimal, where t 0 and t f are the time passing through the starting point and the ending point, respectively; at this time, the planning time needs to be reduced to achieve the optimal time at the same time. And meet the actuator specification; 情形2：第三步得到的规划轨迹的角速度
Figure FDA0002914706090000045
和角加速度
Figure FDA0002914706090000046
虽然大于执行器规范，即关节可执行的最大角速度
Figure FDA0002914706090000047
和角加速度
Figure FDA0002914706090000048
此时，最简单的解决方式就是增加规划时间以达到时间最优同时又满足执行器规范；Case 2: The angular velocity of the planned trajectory obtained in the third step
Figure FDA0002914706090000045
and angular acceleration
Figure FDA0002914706090000046
Although greater than the actuator specification, the maximum angular velocity that the joint can perform
Figure FDA0002914706090000047
and angular acceleration
Figure FDA0002914706090000048
At this point, the simplest solution is to increase the planning time to achieve the optimal time while meeting the actuator specification; 根据以上两种情形，结合比例-积分-微分(PID)控制原理，可设计没有迭代的、简单易实现的、基于条件比例控制的时间最有控制器：According to the above two situations, combined with the proportional-integral-derivative (PID) control principle, a simple and easy-to-implement time-optimized controller based on conditional proportional control without iteration can be designed:
Figure FDA0002914706090000049
Figure FDA0002914706090000049
 其中
Figure FDA00029147060900000410
分别为时间最优控制器中速度和加速度的误差的调节参数，类似PID控制原理中的比例参数；式(7)中的比例控制是否被激活取决于其对应的执行器规范是否得到满足，即条件比例控制的由来；特别地，
Figure FDA00029147060900000411
和
Figure FDA00029147060900000412
可以由各关节或各轴单独定义；在理想条件下，规划时间T＝tf-t0达到最优时，规划轨迹的角速度和角加速度与执行器规范允许的最大角速度和角加速度相等，即
Figure FDA0002914706090000051
此外，根据多约束和多目标的需求，式(7)右式需要加入最大加加速度项，其他运动学和动力学约束项；为了提高时间最优控制器的瞬态过渡性能，也需要参考PID控制加入积分控制和微分控制；in
Figure FDA00029147060900000410
are the adjustment parameters of the error of speed and acceleration in the time-optimized controller, which are similar to the proportional parameters in the PID control principle; whether the proportional control in formula (7) is activated depends on whether the corresponding actuator specifications are satisfied, that is, The origin of conditional proportional control; in particular,
Figure FDA00029147060900000411
and
Figure FDA00029147060900000412
It can be defined independently by each joint or each axis; under ideal conditions, when the planning time T=t f -t 0 reaches the optimum, the angular velocity and angular acceleration of the planned trajectory are equal to the maximum angular velocity and angular acceleration allowed by the actuator specification, that is
Figure FDA0002914706090000051
In addition, according to the requirements of multi-constraints and multi-objectives, the maximum jerk term and other kinematic and dynamic constraint terms need to be added to the right side of equation (7). In order to improve the transient transition performance of the time-optimized controller, it is also necessary to refer to the PID The control adds integral control and differential control; 该算法必须先讨论情形1在讨论情形2以保证执行器规范等约束完全得到满足；首先，初始化规划参数：规划起始点p0，规划终止点pf，规划运动时间T＝tf-t0；其次，基于七次多项式插值规划轨迹：
Figure FDA0002914706090000052
再次利用得到的逆解求出各关节角度：
Figure FDA0002914706090000053
最后判定是否满足情形1：如果满足，则以式(7)来减小规划时间T＝tf-t0重新规划轨迹；如果不满足，则进入情形2；在情形2中，与情形1相同，首先基于七次多项式插值规划轨迹：
Figure FDA0002914706090000054
再次利用得到的逆解求出各关节角度：
Figure FDA0002914706090000055
最后判定是否满足情形2：如果满足，则以式(7)来增大规划时间T＝tf-t0重新规划轨迹；如果不满足，则退出情形2；两种情形讨论结束；The algorithm must first discuss case 1 and discuss case 2 to ensure that constraints such as actuator specifications are fully satisfied; first, initialize the planning parameters: planning start point p 0 , planning end point p f , planning motion time T=t f -t 0 ; Second, the trajectory is planned based on the seventh-order polynomial interpolation:
Figure FDA0002914706090000052
Use the obtained inverse solution again to find the joint angles:
Figure FDA0002914706090000053
Finally, determine whether the situation 1 is satisfied: if it is satisfied, reduce the planning time T=t f -t 0 to re-plan the trajectory according to formula (7); if not, then enter the situation 2; in the situation 2, it is the same as the situation 1 , first plan the trajectory based on the seventh degree polynomial interpolation:
Figure FDA0002914706090000054
Use the obtained inverse solution again to find the joint angles:
Figure FDA0002914706090000055
Finally, it is determined whether the situation 2 is satisfied: if it is satisfied, the planning time T = t f -t 0 is increased by formula (7) to re-plan the trajectory; if it is not satisfied, the situation 2 is exited; the discussion of the two situations ends; ③输出
Figure FDA0002914706090000056
作为关节控制的给定，算法结束。③Output
Figure FDA0002914706090000056
Given the joint control, the algorithm ends. 2.权利要求1所述的一种机械臂逆解多解选择和时间最优轨迹规划算法，其特征在于：该算法中，多项式插值轨迹规划的阶次由具体的轨迹规划需求决定，不仅限于七次；在关节空间内，在连续轨迹规划中，该算法可以得到同样的效果；该算法也适用于其他逆解存在多解的多自由度机械臂或串联机器人。2. The multi-solution selection and time-optimal trajectory planning algorithm of a robotic arm inverse solution according to claim 1, characterized in that: in the algorithm, the order of the polynomial interpolation trajectory planning is determined by the specific trajectory planning requirements, not limited to Seven times; in the joint space, in the continuous trajectory planning, the algorithm can get the same effect; the algorithm is also suitable for other multi-degree-of-freedom manipulators or tandem robots with multiple solutions in other inverse solutions. 3.权利要求1所述的一种机械臂逆解多解选择和时间最优轨迹规划算法，其特征在于：本算法没有采用迭代原理，其算法架构简洁高效、具备快速从逆解多解中选择最合适的一组解并规划时间最优轨迹的能力，可以根据当前的机械臂位姿迅速从逆解多解中选择最合适的一组解，有效规避奇异位型，实现不同逆解之间的平滑切换，保证机械臂控制和动作的连续性和不振荡；并利用七次多项式和条件比例控制相结合的轨迹规划方法规划出时间最优的轨迹，保证关节角度、角速度、角加速度、角加加速度的平滑连续可导，同时满足多目标多约束条件，满足执行器规范，如关节角速度最大值约束或关节角加速度最大值约束等；该算法大大减小了多解选择和轨迹规划的计算量，可以实现实时高效的时间最优轨迹规划，并可推广到低算力控制器上应用。3. The multi-solution selection and time-optimal trajectory planning algorithm for a robotic arm inverse solution according to claim 1, characterized in that: this algorithm does not adopt the iterative principle, and its algorithm architecture is concise and efficient, and has the ability to quickly recover from the inverse solution and multi-solution. The ability to select the most suitable set of solutions and plan the optimal trajectory of time, can quickly select the most suitable set of solutions from multiple inverse solutions according to the current position of the manipulator, effectively avoid singular positions, and realize the integration of different inverse solutions. The smooth switching between the control and action of the manipulator ensures the continuity and non-oscillation of the control and action of the manipulator; and uses the trajectory planning method combining the seven-order polynomial and the conditional proportional control to plan the time-optimized trajectory to ensure the joint angle, angular velocity, angular acceleration, The angular jerk is smooth and continuously derivable, and at the same time, it satisfies the multi-objective and multi-constraint conditions, and meets the actuator specifications, such as the maximum joint angular velocity constraint or the maximum joint angular acceleration constraint; this algorithm greatly reduces the multi-solution selection and trajectory planning. It can realize real-time and efficient time optimal trajectory planning, and can be extended to low computing power controllers.
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