1、中南大学 蔡自兴，谢 斌 zxcai, 2010,机器人学基础 第七章 机器人轨迹规划,1,Ch.7 Trajectory Planning of Robots,Fundamentals of Robotics,Ch.7 Trajectory Planning of Robots,2,Ch.7 Trajectory Planning of Robots,Ch.7 Trajectory Planning of Robots,3,Ch.7 Trajectory Planning of Robots,7.1 General Considerations in Trajectory Planning
2、 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).,4,7.1 General considerations,7.1 General Considerations in Trajectory Planning,Trajectory : Time history of position, velocity and acceleration for eac
3、h DOF Path points : Initial, final and via points Constraints: Spatial, time, smoothness,5,7.1 General considerations,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 Cartesian space W
4、e 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,6,General Considerations - Solution Space,7.1 General considerations,Cartesian planning difficulties:,7,General C
5、onsiderations - Solution Space,7.1 General considerations,Initial (A) and Goal (B) Points are reachable, but intermediate points (C) unreachable.,Ch.7 Trajectory Planning of Robots,8,Ch.7 Trajectory Planning of Robots,Joint-Space Schemes Each path point is “converted“ into a set of desired joint ang
6、les 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 in
7、dependent with other joints.,9,7.2 Interpolated Calculation of Joint Trajectories 关节轨迹的插值计算,7.2 JointSpace Schemes,Choice of interpolation function is not unique!,10,Joint-Space Schemes,7.2 JointSpace Schemes,Several possible path shapes for a single joint.,Some possible interpolation functions: Cub
8、ic 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,11,Joint-Space Schemes,7.2 JointSpace Schemes,In making a single smooth motion, at least four constraints on are
9、 evident:,12,7.2.1 Cubic Polynomials 三次多项式插值,7.2 JointSpace Schemes,Combining the four constraints yields four equations with four unknowns:,13,7.2.1 Cubic Polynomials,7.2 JointSpace Schemes,These four constraints uniquely specify a particular cubic:,14,7.2.1 Cubic Polynomials,The joint velocity and
10、 acceleration along this path are:,7.2 JointSpace Schemes,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 man
11、ipulator to rest at the goal. Plot the position, velocity, and acceleration of the joint as a function of time.,15,7.2.1 Cubic Polynomials,7.2 JointSpace Schemes,Solution: Plugging 0 =15，f =75，tf = 3 into (7.6), we find,16,7.2.1 Cubic Polynomials,7.2 JointSpace Schemes,Solution:,17,7.2.1 Cubic Polyn
12、omials,7.2 JointSpace Schemes,Starts at 15 degrees and ends at 75 degrees!,Solution:,18,7.2.1 Cubic Polynomials,7.2 JointSpace Schemes,Starts and ends at rest!,Solution:,19,7.2.1 Cubic Polynomials,7.2 JointSpace Schemes,Acceleration profile is linear!,If we come to rest at each pointuse formula from
13、 previous slide or continuous motion (no stops)need velocities at intermediate points: Initial Conditions:,20,7.2.2 Cubic polynomials with via points 过路径点的三次多项式插值,7.2 JointSpace Schemes,Solutions:,How to specify velocity at the via points: The user specifies the desired velocity at each via point in
14、 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
15、at the via points in such a way as to cause the acceleration at the via points to be continuous.,21,7.2 JointSpace Schemes,7.2.2 Cubic polynomials with via points,Higher order polynomials are sometimes used for path segments. For example, if we wish to be able to specify the position, velocity, and
16、acceleration at the beginning and end of a path segment, a quintic polynomial is required:,22,7.2.3 Higher-order polynomials 高阶多项式插值,7.2 JointSpace Schemes,Where the constraints are given as:,23,7.2.3 Higher-order polynomials,7.2 JointSpace Schemes,Solution to these equations:,24,7.2.3 Higher-order
17、polynomials,7.2 JointSpace Schemes,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.,25,7.2.4 Linear function with parabolic blends 用抛物线过渡的线性插值,7.2 JointSpace Schemes,Disconti
18、nuous 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.,26,7.2.4 Linear function wi
19、th parabolic blends,7.2 JointSpace Schemes,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.,27,7.2.4 Linear function with par
20、abolic blends,7.2 JointSpace Schemes,The velocity at the end of the blend region must equal the velocity of the linear section:,28,7.2.4 Linear function with parabolic blends,7.2 JointSpace Schemes,Let t=2th ，combining (7.13) and (7.14),29,7.2.4 Linear function with parabolic blends,The acceleration
21、 chosen must be sufficiently high, to ensure the existence of a solution:,7.2 JointSpace Schemes,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.,30,7.2.5 Linear function with parabolic
22、blends for a path with via points,过路径点的用抛物线过渡的线性插值,7.2 JointSpace Schemes,Multi-segment linear path with blends.,Given: positions desired time durations the magnitudes of the accelerations Compute: blends times straight segment times slopes (velocities) signed accelerations,31,7.2 JointSpace Schemes
23、,7.2.5 Linear function with parabolic blends for a path with via points,Inside segment:,32,7.2 JointSpace Schemes,7.2.5 Linear function with parabolic blends for a path with via points,First segment:,33,7.2 JointSpace Schemes,7.2.5 Linear function with parabolic blends for a path with via points,Las
24、t segment:,34,7.2 JointSpace Schemes,7.2.5 Linear function with parabolic blends for a path with via points,To go through the actual via points: Introduce “Pseudo Via Points” Use sufficiently high acceleration,35,7.2 JointSpace Schemes,7.2.5 Linear function with parabolic blends for a path with via
25、points,Ch.7 Trajectory Planning of Robots,36,Ch.7 Trajectory Planning of Robots,When 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 circul
26、ar, 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.,7.3 Cartesian-Space Schemes,37,7.3 Cartesian-Space Schemes,Description of a task,7.3 Cartesian-Space Schem
27、es,38,7.3 Cartesian-Space Schemes,Cartesian straight line motionMove from point Pi to Pi+1 , which described by relative homogenous transformation:,7.3 Cartesian-Space Schemes,39,7.3 Cartesian-Space Schemes,In order to ensure continuous velocities in trajectory, a spline of linear functions with par
28、abolic 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.,7.3 Cartesian-Space Schemes,40,7.3 Cartesian-Space Schemes,Cartesian planning difficulties (1/3):,41,In
29、itial (A) and Goal (B) Points are reachable, but intermediate points (C) unreachable.,7.3 Cartesian-Space Schemes,7.3 Cartesian-Space Schemes,42,Approaching singularities some joint velocities go to causing deviation from the path.,7.3 Cartesian-Space Schemes,7.3 Cartesian-Space Schemes,Cartesian pl
30、anning difficulties (2/3):,43,Start point (A) and goal point (B) are reachable in different joint space solutions (The middle points are reachable from below.),7.3 Cartesian-Space Schemes,7.3 Cartesian-Space Schemes,Cartesian planning difficulties (3/3):,Ch.7 Trajectory Planning of Robots,44,Ch.7 Tr
31、ajectory Planning of Robots,7.4 Path Generation at Real-Time,At 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.,7.4 Path Generation
32、 at Run Time,45,7.4.1 Generation of joint space paths,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.,7.4 Path
33、Generation at Run Time,46,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 segment. In the linear portion, the trajectory for each joint is calculated as,7.4 Path Gener
34、ation at Run Time,47,7.4.1 Generation of joint space paths,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 segment. In the blend region, the trajectory for each joint
35、is calculated as,7.4 Path Generation at Run Time,48,7.4.1 Generation of joint space paths,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 portion of the segme
36、nt, each degree of freedom in X is calcuated as,7.4 Path Generation at Run Time,49,7.4.2 Generation of Cartesian space paths,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.
37、 In the blend region, the trajectory for each degree of freedom is calculated as,7.4 Path Generation at Run Time,50,7.4.2 Generation of Cartesian space paths,Finally, this Cartesian trajectory ( ) must be converted into equivalent joint space quantities. A complete analytical solution to this proble
38、m would use: inverse kinematics to calculate joint positions, inverse Jacobian for velocities, inverse Jacobian plus its derivative for accelerations.,7.4 Path Generation at Run Time,51,7.4.2 Generation of Cartesian space paths,52,7.5 Summary 小结,General Considerations in Robot Trajectory Planning Jo
39、int-Space Schemes 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 Cartesian-Space Schemes Track of any desired shape More expensive at run time Discontinuity problems Real Time Generation of Planning Trajectories,7.5 Summary,Thank you For Attention!,Fundamentals of Robotics,53,
