机器人学基础_第7章_机器人轨迹规划分析解析

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7.2.1 Cubic Polynomials
Solution: (t ) a0 a1t a2t 2 a3t 3 15 20t 2 4.44t 3
Starts at 15 degrees and ends at 75 degrees!
7.2 Joint–Space Schemes
(0) 0
(t f ) 0
(7.2)
7.2 Joint–Space Schemes
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7.2.1 Cubic Polynomials
Combining the four constraints yields four equations with four unknowns: 0 a0 2 3 f a0 a1t f a2t f a3t f (7.5) 0 a1 0 a1 2a2t f 3a3t 2 f a0 0
3 a2 2 ( f 0 ) tf 2 a3 3 ( f 0 ) tf a1 0
(7.6)
7.2 Joint–Space Schemes
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7.2.1 Cubic Polynomials
These four constraints uniquely specify a particular cubic:
(t ) a0 a1t a2t 2 a3t 3
(7.3)
The joint velocity and acceleration along this path are: (t ) a1 2a 2 t 3a3t 2 (7.4) (t ) 2a 2 6a3t
Joint-Space Schemes
Each path point is "converted" into a set of desired joint angles by application of the inverse kinematics.
A smooth function is found for each of the n joints which pass through the via points and end at the goal point. Time required for each segment is the same for each joint. The determination of the desired joint angle function for a particular joint is independent with other joints.
7.3 Planning of Cartesian Path Trajectories
7.4 7.5 Real Time Generation of Planning Trajectories Summary
Ch.7 Trajectory Planning of Robots
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7.2 Interpolated Calculation of Joint Trajectories 关节轨迹的插值计算
Basic Problem: Move the manipulator arm from some initial position to some desired final position (May be going through some via points).
7.1 General considerations
7.2 Joint–Space Schemes
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7.2.1 Cubic Polynomials 三次多项式插值
In making a single smooth motion, at least four constraints on t are evident: ( 0) 0 (t f ) f (7.1)
Solution:
(t ) 2a2 6a3t 40 26.66t
Βιβλιοθήκη Baidu
Acceleration profile is linear!
7.2 Joint–Space Schemes
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7.2.2 Cubic polynomials with via points 过路径点的三次多项式插值
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7.1 General Considerations in Trajectory Planning Trajectory : Time history of position, velocity and acceleration for each DOF Path points : Initial, final and via points Constraints: Spatial, time, smoothness
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Joint-Space Schemes Some possible interpolation functions:
Cubic polynomials
Cubic polynomials for a path with via points Higher-order polynomials Linear function with parabolic blends Linear function with parabolic blends for a path with via points
7.3 Planning of Cartesian Path Trajectories
7.4 7.5 Real Time Generation of Planning Trajectories Summary
Ch.7 Trajectory Planning of Robots
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7.1 General Considerations in Trajectory Planning 轨迹规划应考虑的问题
7.2 Joint–Space Schemes
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7.2.1 Cubic Polynomials
Solution: Plugging θ0 =15，θf =75，tf = 3 into (7.6), we find
a0 0 15 a1 0 3 a2 2 ( f 0 ) 20 tf 2 a3 3 ( f 0 ) 4.44 tf
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7.2.1 Cubic Polynomials
Solution:
(t ) a1 2a2t 3a3t 2 40t 13.33t 2
Starts and ends at rest!
7.2 Joint–Space Schemes
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7.2.1 Cubic Polynomials
Ch.7 Trajectory Planning of Robots
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Ch.7 Trajectory Planning of Robots
7.1 7.2
General Considerations in Robot Trajectory Planning Interpolated Calculation of Joint Trajectories
7.1 7.2 7.3 7.4 7.5 General Considerations in Robot Trajectory Planning Interpolated Calculation of Joint Trajectories Planning of Cartesian Path Trajectories Real Time Generation of Planning Trajectories Summary
机器人学基础
第七章 机器人轨迹规划
Fundamentals of Robotics
Ch.7 Trajectory Planning of Robots
中南大学 蔡自兴，谢 斌 zxcai, xiebin@mail.csu.edu.cn 2010
Fundamentals of Robotics
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Ch.7 Trajectory Planning of Robots
7.1 General considerations
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Ch.7 Trajectory Planning of Robots
7.1 7.2 General Considerations in Robot Trajectory Planning Interpolated Calculation of Joint Trajectories
7.2 Joint–Space Schemes
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Joint-Space Schemes Choice of interpolation function is not unique!
Several possible path shapes for a single joint.
7.2 Joint–Space Schemes
7.2 Joint–Space Schemes
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7.2.1 Cubic Polynomials
Eg. 7.1 A single-link robot with a rotary joint is motionless at θ = 15 degrees. It is desired to move the joint in a smooth manner to θ= 75 degrees in 3 seconds. Find the coefficients of a cubic which accomplishes this motion and brings the manipulator to rest at the goal. Plot the position, velocity, and acceleration of the joint as a function of time.
(t ) a0 a1t a2t 2 a3t 3 15 20t 2 4.44t 3 (t ) a1 2a2t 3a3t 2 40t 13.33t 2 (t ) 2a2 6a3t 40 26.66t
7.2 Joint–Space Schemes
7.1 General considerations
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General Considerations - Solution Space Cartesian planning difficulties:
Initial (A) and Goal (B) Points are reachable, but intermediate points (C) unreachable.
Cartesian space
We can track a shape (for orientation : equivalent axes, Euler angles,…) More expensive at run time (after the path is calculated need joint angles in a lot of points) Discontinuity problems
7.1 General considerations
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General Considerations - Solution Space
Joint space
Easy to go through via points
(Solve inverse kinematics at all path points) No problems with singularities Less calculations Can not follow straight line