机器人学基础-第7章-机器人轨迹规划-蔡自兴.ppt

1、中南大学中南大学蔡自兴，谢蔡自兴，谢 斌斌zxcai,2010机器人学基础机器人学基础第七章第七章 机器人轨迹规划机器人轨迹规划1Ch.7 Trajectory Planning of RobotsFundamentals of RoboticsCh.7 Trajectory Planning of Robots 2Ch.7 Trajectory Planning of Robots7.1General Considerations in Robot Trajectory Planning7.2Interpolated Calculation of Joint Trajectories 7.3

2、Planning of Cartesian Path Trajectories7.4Real Time Generation of Planning Trajectories7.5SummaryCh.7 Trajectory Planning of Robots 37.1General Considerations in Robot Trajectory Planning7.2Interpolated Calculation of Joint Trajectories 7.3Planning of Cartesian Path Trajectories7.4Real Time Generati

3、on of Planning Trajectories7.5SummaryCh.7 Trajectory Planning of Robots7.1 General Considerations in Trajectory Planning 轨迹规划应考虑的问题轨迹规划应考虑的问题Basic Problem:Move the manipulator arm from some initial position to some desired final position(May be going through some via points).47.1 General considerati

4、ons7.1 General Considerations in Trajectory PlanningTrajectory:Time history of position,velocity and acceleration for each DOFPath points:Initial,final and via pointsConstraints:Spatial,time,smoothness57.1 General considerationsJoint spaceEasy to go through via points(Solve inverse kinematics at all

5、 path points)No problems with singularitiesLess calculationsCan not follow straight lineCartesian spaceWe 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 problems6General Consid

6、erations-Solution Space7.1 General considerationsCartesian planning difficulties:7General Considerations-Solution Space7.1 General considerationsInitial(A)and Goal(B)Points are reachable,but intermediate points(C)unreachable.Ch.7 Trajectory Planning of Robots 87.1General Considerations in Robot Traj

7、ectory Planning7.2Interpolated Calculation of Joint Trajectories 7.3Planning of Cartesian Path Trajectories7.4Real Time Generation of Planning Trajectories7.5SummaryCh.7 Trajectory Planning of RobotsJoint-Space SchemesEach path point is converted into a set of desired joint angles by application of

8、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 join

9、ts.97.2 Interpolated Calculation of Joint Trajectories 关节轨迹的插值计算关节轨迹的插值计算7.2 JointSpace SchemesChoice of interpolation function is not unique!10Joint-Space Schemes 7.2 JointSpace SchemesSeveral possible path shapes for a single joint.Some possible interpolation functions:Cubic polynomials Cubic poly

10、nomials for a path with via pointsHigher-order polynomials Linear function with parabolic blendsLinear function with parabolic blends for a path with via points11Joint-Space Schemes 7.2 JointSpace SchemesIn making a single smooth motion,at least four constraints on are evident:127.2.1 Cubic Polynomi

11、als 三次多项式插值三次多项式插值7.2 JointSpace SchemesCombining the four constraints yields four equations with four unknowns:137.2.1 Cubic Polynomials7.2 JointSpace SchemesThese four constraints uniquely specify a particular cubic:147.2.1 Cubic PolynomialsThe joint velocity and acceleration along this path are:7

12、.2 JointSpace SchemesEg.7.1 A single-link robot with a rotary joint is motionless at =15 degrees.It is desired to move the joint in a smooth manner to=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 posit

13、ion,velocity,and acceleration of the joint as a function of time.157.2.1 Cubic Polynomials7.2 JointSpace SchemesSolution:Plugging 0=15，f=75，tf=3 into(7.6),we find167.2.1 Cubic Polynomials7.2 JointSpace SchemesSolution:177.2.1 Cubic Polynomials7.2 JointSpace SchemesStarts at 15 degrees and ends at 75

14、 degrees!Solution:187.2.1 Cubic Polynomials7.2 JointSpace SchemesStarts and ends at rest!Solution:197.2.1 Cubic Polynomials7.2 JointSpace SchemesAcceleration profile is linear!If we come to rest at each pointuse formula from previous slideor continuous motion(no stops)need velocities at intermediate

15、 points:Initial Conditions:207.2.2 Cubic polynomials with via points 过路径点的三次多项式插值过路径点的三次多项式插值7.2 JointSpace SchemesSolutions:How to specify velocity at the via points:The user specifies the desired velocity at each via point in terms of a Cartesian linear and angular velocity of the tool frame at th

16、at instant.The system automatically chooses the velocities at the via points by applying a suitable heuristic in either Cartesian space or joint space(average of 2 sides etc.).The system automatically chooses the velocities at the via points in such a way as to cause the acceleration at the via points to be continuous.217.2 JointSpace Schemes7.2.2 Cubic polynomials with via pointsHigher order polynomials are sometimes used for path segments.For example,if we wish to be able to specify the positi