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1、Ch.7 Trajectory Planning of Robots 1Ch.7 Trajectory Planning of Robots7.1General Considerations in Robot Trajectory Planning7.2Interpolated Calculation of Joint Trajectories 7.3Planning of Cartesian Path Trajectories7.4Real Time Generation of Planning Trajectories7.5Summary第1页/共53页Ch.7 Trajectory Pl
2、anning of Robots 27.1General Considerations in Robot Trajectory Planning7.2Interpolated Calculation of Joint Trajectories 7.3Planning of Cartesian Path Trajectories7.4Real Time Generation of Planning Trajectories7.5SummaryCh.7 Trajectory Planning of Robots第2页/共53页7.1 General Considerations in Trajec
3、tory Planning 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).37.1 General considerations第3页/共53页7.1 General Considerations in Trajectory PlanningTrajectory:Time history of position,velocity and accelera
4、tion for each DOFPath points:Initial,final and via pointsConstraints:Spatial,time,smoothness47.1 General considerations第4页/共53页Joint spaceEasy to go through via points(Solve inverse kinematics at all path points)No problems with singularitiesLess calculationsCan not follow straight lineCartesian spa
5、ceWe 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 problems5General Considerations-Solution Space7.1 General considerations第5页/共53页Cartesian planning difficulties:6General Con
6、siderations-Solution Space7.1 General considerationsInitial(A)and Goal(B)Points are reachable,but intermediate points(C)unreachable.第6页/共53页Ch.7 Trajectory Planning of Robots 77.1General Considerations in Robot Trajectory Planning7.2Interpolated Calculation of Joint Trajectories 7.3Planning of Carte
7、sian Path Trajectories7.4Real Time Generation of Planning Trajectories7.5SummaryCh.7 Trajectory Planning of Robots第7页/共53页Joint-Space SchemesEach 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 wh
8、ich 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.87.2 Interpolated Calculation of Joint Trajectories 关节轨迹的插值计算7.2 JointSpac
9、e Schemes第8页/共53页Choice of interpolation function is not unique!9Joint-Space Schemes 7.2 JointSpace SchemesSeveral possible path shapes for a single joint.第9页/共53页Some possible interpolation functions:Cubic polynomials Cubic polynomials for a path with via pointsHigher-order polynomials Linear funct
10、ion with parabolic blendsLinear function with parabolic blends for a path with via points10Joint-Space Schemes 7.2 JointSpace Schemes第10页/共53页In making a single smooth motion,at least four constraints on are evident:117.2.1 Cubic Polynomials 三次多项式插值三次多项式插值7.2 JointSpace Schemes第11页/共53页Combining the
11、 four constraints yields four equations with four unknowns:127.2.1 Cubic Polynomials7.2 JointSpace Schemes第12页/共53页These four constraints uniquely specify a particular cubic:137.2.1 Cubic PolynomialsThe joint velocity and acceleration along this path are:7.2 JointSpace Schemes第13页/共53页Eg.7.1 A singl
12、e-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 position,velocity,and acceleration of th
13、e joint as a function of time.147.2.1 Cubic Polynomials7.2 JointSpace Schemes第14页/共53页Solution:Plugging 0=15，f=75，tf=3 into(7.6),we find157.2.1 Cubic Polynomials7.2 JointSpace Schemes第15页/共53页Solution:167.2.1 Cubic Polynomials7.2 JointSpace SchemesStarts at 15 degrees and ends at 75 degrees!第16页/共53
14、页Solution:177.2.1 Cubic Polynomials7.2 JointSpace SchemesStarts and ends at rest!第17页/共53页Solution:187.2.1 Cubic Polynomials7.2 JointSpace SchemesAcceleration profile is linear!第18页/共53页If we come to rest at each pointuse formula from previous slideor continuous motion(no stops)need velocities at in
15、termediate points:Initial Conditions:197.2.2 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:第19页/共53页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 t
16、he tool frame at that 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 accelerat
17、ion at the via points to be continuous.207.2 JointSpace Schemes7.2.2 Cubic polynomials with via points第20页/共53页Higher order polynomials are sometimes used for path segments.For example,if we wish to be able to specify the position,velocity,and acceleration at the beginning and end of a path segment,
18、a quintic polynomial is required:217.2.3 Higher-order polynomials高阶多项式插值高阶多项式插值7.2 JointSpace Schemes第21页/共53页Where the constraints are given as:227.2.3 Higher-order polynomials7.2 JointSpace Schemes第22页/共53页Solution to these equations:237.2.3 Higher-order polynomials7.2 JointSpace Schemes第23页/共53页L
19、inear interpolation(Straight line):Note:Although the motion of each joint in this scheme is linear,the end-effector in general does not move in a straight line in space.247.2.4 Linear function with parabolic blends 用抛物线过渡的线性插值用抛物线过渡的线性插值7.2 JointSpace SchemesDiscontinuous velocity-can not be control
20、led!第24页/共53页To create a smooth path with continous position and velocity,we start with the linear function but add a parabolic blend region at each path point.Constant acceleration is used during the blend portion to change velocity smoothly.257.2.4 Linear function with parabolic blends7.2 JointSpa
21、ce Schemes第25页/共53页Assume that the parabolic blends both have the same duration,and therefore the same constant acceleration(modulo a sign).There are many solutions to the problem-but the answer is always symmetric about the halfway point.267.2.4 Linear function with parabolic blends7.2 JointSpace S
22、chemes第26页/共53页The velocity at the end of the blend region must equal the velocity of the linear section:277.2.4 Linear function with parabolic blends7.2 JointSpace Schemes第27页/共53页Let t=2th，combining(7.13)and(7.14)287.2.4 Linear function with parabolic blendsThe acceleration chosen must be sufficie
23、ntly high,to ensure the existence of a solution:7.2 JointSpace Schemes第28页/共53页Below shows a set of joint space via points for some joints.Linear functions connect the via points,and parabolic blend regions are added around each via point.297.2.5 Linear function with parabolic blendsfor a path with
24、via points过路径点的用抛物线过渡的线性插值过路径点的用抛物线过渡的线性插值7.2 JointSpace SchemesMulti-segment linear path with blends.第29页/共53页Given:positionsdesired time durations the magnitudes of the accelerationsCompute:blends timesstraight segment times slopes(velocities)signed accelerations307.2 JointSpace Schemes7.2.5 Linea
25、r function with parabolic blendsfor a path with via points第30页/共53页Inside segment:317.2 JointSpace Schemes7.2.5 Linear function with parabolic blendsfor a path with via points第31页/共53页First segment:327.2 JointSpace Schemes7.2.5 Linear function with parabolic blendsfor a path with via points第32页/共53页
26、Last segment:337.2 JointSpace Schemes7.2.5 Linear function with parabolic blendsfor a path with via points第33页/共53页To go through the actual via points:Introduce“Pseudo Via Points”Use sufficiently high acceleration347.2 JointSpace Schemes7.2.5 Linear function with parabolic blendsfor a path with via
27、points第34页/共53页Ch.7 Trajectory Planning of Robots 357.1General Considerations in Robot Trajectory Planning7.2Interpolated Calculation of Joint Trajectories 7.3Planning of Cartesian Path Trajectories7.4Real Time Generation of Planning Trajectories7.5SummaryCh.7 Trajectory Planning of Robots第35页/共53页W
28、hen path shapes are described in terms of functions of Cartesian position and orientation,we can also specify the spatial shape of the path between path points.The most common path shape is a straight line;but circular,sinusoidal,or other path shapes could be used.Cartesian schemes are more computat
29、ionally expensive to execute since at run time,inverse kinematics must be solved at the path update rate.7.3 Cartesian-Space Schemes367.3 Cartesian-Space Schemes第36页/共53页Description of a task7.3 Cartesian-Space Schemes377.3 Cartesian-Space Schemes第37页/共53页Cartesian straight line motionMove from poin
30、t Pi to Pi+1,which described by relative homogenous transformation:7.3 Cartesian-Space Schemes387.3 Cartesian-Space Schemes第38页/共53页In order to ensure continuous velocities in trajectory,a spline of linear functions with parabolic blends is always used.During the linear portion of each segment,since
31、 all three components of position change in a linear fashion,the end-effector will move along a linear path in space.7.3 Cartesian-Space Schemes397.3 Cartesian-Space Schemes第39页/共53页Cartesian planning difficulties(1/3):40Initial(A)and Goal(B)Points are reachable,but intermediate points(C)unreachable
32、.7.3 Cartesian-Space Schemes7.3 Cartesian-Space Schemes第40页/共53页41Approaching singularities some joint velocities go to causing deviation from the path.7.3 Cartesian-Space Schemes7.3 Cartesian-Space SchemesCartesian planning difficulties(2/3):第41页/共53页42Start point(A)and goal point(B)are reachable i
33、n different joint space solutions(The middle points are reachable from below.)7.3 Cartesian-Space Schemes7.3 Cartesian-Space SchemesCartesian planning difficulties(3/3):第42页/共53页Ch.7 Trajectory Planning of Robots 437.1General Considerations in Robot Trajectory Planning7.2Interpolated Calculation of
34、Joint Trajectories 7.3Planning of Cartesian Path Trajectories7.4Real Time Generation of Planning Trajectories7.5SummaryCh.7 Trajectory Planning of Robots第43页/共53页7.4 Path Generation at Real-TimeAt run time the path generator routine constructs the trajectory,usually in terms of ,and feeds this infor
35、mation to the manipulators control system.This path generator computes the trajectory at the path update rate.7.4 Path Generation at Run Time44第44页/共53页In the case of cubic splines,the path generator simply computes(7.3)and(7.4)as t is advanced.When the end of one segment is reached,a new set of cub
36、ic coefficients is recalled,t is set back to zero,and the generation continues.7.4 Path Generation at Run Time45第45页/共53页In the case of linear splines with parabolic blends,the value of time,t,is checked on each update to determine whether we are currently in the linear or the blend portion of the s
37、egment.In the linear portion,the trajectory for each joint is calculated as7.4 Path Generation at Run Time46第46页/共53页In the case of linear splines with parabolic blends,the value of time,t,is checked on each update to determine whether we are currently in the linear or the blend portion of the segme
38、nt.In the blend region,the trajectory for each joint is calculated as7.4 Path Generation at Run Time47第47页/共53页In the case of linear spline with parabolic blends path.Rewrite(7.45)and(7.46)with the symbol X representing a component of the Cartesian position and orientation vector.In the linear porti
39、on of the segment,each degree of freedom in X is calcuated as7.4 Path Generation at Run Time48第48页/共53页In the case of linear spline with parabolic blends path.Rewrite(7.45)and(7.46)with the symbol X representing a component of the Cartesian position and orientation vector.In the blend region,the tra
40、jectory for each degree of freedom is calculated as7.4 Path Generation at Run Time49第49页/共53页Finally,this Cartesian trajectory()must be converted into equivalent joint space quantities.A complete analytical solution to this problem would use:inverse kinematics to calculate joint positions,inverse Ja
41、cobian for velocities,inverse Jacobian plus its derivative for accelerations.7.4 Path Generation at Run Time50第50页/共53页517.5 Summary 小结General Considerations in Robot Trajectory PlanningJoint-Space SchemesCubic polynomials Cubic polynomials for a path with via pointsHigher-order polynomials Linear f
42、unction with parabolic blendsLinear function with parabolic blends for a path with via pointsCartesian-Space SchemesTrack of any desired shapeMore expensive at run timeDiscontinuity problemsReal Time Generation of Planning Trajectories7.5 Summary第51页/共53页Thank youFor Attention!Fundamentals of Robotics52第52页/共53页感谢您的观看。第53页/共53页