第7章-轨迹规划课件

第6章 机器人轨迹规划6.1General Considerations in Robot Trajectory Planning6.2Interpolated Calculation of Joint Trajectories 6.3 Planning of Cartesian Path Trajectories6.4Real Time Generation of Planning Trajectories6.5 Summary6.1 General Considerations in Trajectory 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).Trajectory:Time history of position,velocity and acceleration for each DOFPath points:Initial,final and via pointsConstraints:Spatial,time,smoothness轨迹规划应考虑的问题Joint spaceEasy to go through via points(Solve inverse kinematics at all path points)No problems with singularitiesLess calculationsCan not follow straight lineGeneral Considerations-Solution SpaceCartesian 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 problemsGeneral Considerations-Solution SpaceCartesian planning difficulties:General Considerations-Solution SpaceInitial(A)and Goal(B)Points are reachable,but intermediate points(C)unreachable.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 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.7.2 Interpolated Calculation of Joint Trajectories 关节轨迹的插值计算Choice of interpolation function is not unique!Joint-Space Schemes Several possible path shapes for a single joint.Some possible interpolation functions:Cubic polynomials Cubic polynomials for a path with via pointsHigher-order polynomials Linear function with parabolic blendsLinear function with parabolic blends for a path with via pointsJoint-Space Schemes In making a single smooth motion,at least four constraints:7.2.1 Cubic Polynomials 三次多项式插值These four constraints uniquely specify a particular cubic:The joint velocity and acceleration along this path are:Cubic PolynomialsCubic Polynomials-4个系数Eg.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 position,velocity,and acceleration of the joint as a function of time.ExampleExample关节运动轨迹-位置Starts at 15 degrees and ends at 75 degrees!Starts and ends at rest!关节运动轨迹-速度Acceleration profile is linear!关节运动轨迹-关节加速度If we come to rest at each pointuse formula from previous slideor continuous motion(no stops)need velocities at intermediate points:Initial Conditions:6.2.2 Cubic polynomials with via points 过路径点的三次多项式插值Desired Velocity at the va pointsThe user specifies the desired velocity at each via point in terms of a Cartesian linear and angular velocity of the 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 acceleration at the via points to be continuous.1 用Jacob矩阵计算如果Jacob矩阵奇异，需用户指定速度用户负担2 启发式方法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,a quintic polynomial is required:6.2.3 Higher-order polynomials高阶多项式插值Where the constraints are given as:Higher-order polynomials高阶多项式插值Solution to these equations:Higher-order polynomials高阶多项式插值Linear 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.6.2.4 Linear function with parabolic blends 用抛物线过渡的线性插值Discontinuous velocity-can not be controlled!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.Linear function with parabolic blendsAssume 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.Linear function with parabolic blendsThe velocity at the end of the blend region must equal the velocity of the linear section:Linear function with parabolic blendsLet t=2th，combining(7.13)and(7.14)The acceleration chosen must be sufficiently high,to ensure the existence of a solution:Linear function with parabolic blendsBelow 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.6.2.5 Linear function with parabolic blendsfor a path with 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Multi-segment linear path with blends.Given:positionsdesired time durations the magnitudes of the accelerationsCompute:blends timesstraight segment times slopes(velocities)signed accelerationsLinear function with parabolic blendsfor a path with via pointsInside segment:Linear function with parabolic blendsfor a path with via pointsFirst segment:Linear function with parabolic blendsfor a path with via pointsLast segment:Linear function with parabolic blendsfor a path with via pointsTo go through the actual via points:Introduce“Pseudo Via Points”Use sufficiently high accelerationLinear function with parabolic blendsfor a path with via pointsWhen 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 computationally expensive to execute since at run time,inverse kinematics must be solved at the path update rate.6.3 Cartesian-Space SchemesDescription of a taskCartesian-Space SchemesCartesian straight line motionMove from point Pi to Pi+1,which described by relative homogenous transformation:Cartesian-Space SchemesIn 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 all three components of position change in a linear fashion,the end-effector will move along a linear path in space.Cartesian-Space SchemesCartesian-Space SchemesInitial(A)and Goal(B)Points are reachable,but intermediate points(C)unreachable.Geometric Problems with Cartesian PathApproaching singularities some joint velocities go to causing deviation from the path.Geometric Problems with Cartesian PathStart point(A)and goal point(B)are reachable in different joint space solutions(The middle points are reachable from below.)Geometric Problems with Cartesian Path6.4 Path Generation at Real-TimeAt run time the path generator routine constructs the trajectory,usually in terms of ,and feeds this information to the manipulators control system.This path generator computes the trajectory at the path update rate.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 cubic coefficients is recalled,t is set back to zero,and the generation continues.6.4.1 Path Generation in Joint SpaceIn 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 segment.In the linear portion,the trajectory for each joint is calculated asPath Generation in Joint SpaceIn 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 segment.In the blend region,the trajectory for each joint is calculated asPath Generation in Joint SpaceIn 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 portion of the segment,each degree of freedom in X is calcuated as6.4.2 Generation of Cartesian space pathsIn 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 trajectory for each degree of freedom is calculated asGeneration of Cartesian space pathsFinally,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 Jacobian for velocities,inverse Jacobian plus its derivative for accelerations.Generation of Cartesian space pathsSummary 小结General Considerations in Robot Trajectory PlanningJoint-Space SchemesCubic polynomials Cubic polynomials for a path with via pointsHigher-order polynomials Linear function 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 Trajectories