機(jī)器人學(xué)基礎(chǔ)機(jī)器人軌跡規(guī)劃（課堂PPT）

1、1中南大學(xué)中南大學(xué)蔡自興，謝蔡自興，謝 斌斌zxcai, 2010機(jī)器人學(xué)基礎(chǔ)機(jī)器人學(xué)基礎(chǔ)第七章第七章 機(jī)器人軌跡規(guī)劃機(jī)器人軌跡規(guī)劃1Ch.7 Trajectory Planning of RobotsFundamentals of Robotics2Ch.7 Trajectory Planning of Robots 2Ch.7 Trajectory Planning of Robots3Ch.7 Trajectory Planning of Robots 3Ch.7 Trajectory Planning of Robots47.1 General Considerations in Tr

2、ajectory Planning 軌跡規(guī)劃應(yīng)考慮的問題軌跡規(guī)劃應(yīng)考慮的問題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 considerations57.1 General Considerations in Trajectory PlanningTrajectory : Time history of position, velocity a

3、nd acceleration for each DOFPath points : Initial, final and via pointsConstraints: Spatial, time, smoothness57.1 General considerations6Joint spaceEasy to go through via points(Solve inverse kinematics at all path points)No problems with singularitiesLess calculationsCan not follow straight lineCar

4、tesian 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 Considerations - Solution Space7.1 General considerations7Cartesian planning difficulties:7Gen

5、eral Considerations - Solution Space7.1 General considerationsInitial (A) and Goal (B) Points are reachable, but intermediate points (C) unreachable.8Ch.7 Trajectory Planning of Robots 8Ch.7 Trajectory Planning of Robots9Joint-Space SchemesEach 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.97.2 Interpolated Calculation of Joint Trajectories 關(guān)節(jié)軌跡的插值計算關(guān)節(jié)軌跡的插值計算7.2 JointSpace Schemes10Choice of interpolation function is not unique!10Joint-Space Schemes 7.2 JointSpace SchemesSeveral possible path shapes for a single joint.11Some possible interpolation functions

8、: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 points11Joint-Space Schemes 7.2 JointSpace Schemes12In making a single smooth motion, at least four constraints on ar

9、e evident:127.2.1 Cubic Polynomials 三次多項式插值三次多項式插值(7.1)(7.2)fft)()0(00)(0)0(ft7.2 JointSpace Schemes t13Combining the four constraints yields four equations with four unknowns:137.2.1 Cubic Polynomials(7.5)7.2 JointSpace Schemes23211332210003200fffffftataaatatataaa00120230303()2()ffffaaatat (7.6)14T

10、hese four constraints uniquely specify a particular cubic: 147.2.1 Cubic Polynomials332210)(tatataatThe joint velocity and acceleration along this path are:taattataat32232162)(32)( (7.4)(7.3)7.2 JointSpace Schemes15Eg. 7.1 A single-link robot with a rotary joint is motionless at = 15 degrees. It is

11、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.157.2.1 Cubic Polynomials7.2 Joi

12、ntSpace Schemes16Solution: Plugging 0 =15，f =75，tf = 3 into (7.6), we find167.2.1 Cubic Polynomials0012023031503()202()4.44ffffaaatat 232301232212323( )15204.44( )234013.33( )264026.66taata ta ttttaa ta ttttaa tt7.2 JointSpace Schemes17Solution:177.2.1 Cubic Polynomials7.2 JointSpace Schemes23230123

13、( )15204.44taa ta ta tttStarts at 15 degrees and ends at 75 degrees!18Solution:187.2.1 Cubic Polynomials7.2 JointSpace Schemes22123( )234013.33taa ta tttStarts and ends at rest!19Solution:197.2.1 Cubic Polynomials7.2 JointSpace Schemes23( )264026.66taa ttAcceleration profile is linear!20If we come t

14、o rest at each pointuse formula from previous slideor continuous motion (no stops)need velocities at intermediate points:Initial Conditions:207.2.2 Cubic polynomials with via points 過路徑點的三次多項式插值過路徑點的三次多項式插值7.2 JointSpace Schemesfft)()0(00010200230032321()21()()fffffffffaaatttatt Solutions:2321103322

15、100032ffffffftataaatatataaa21How 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 that instant.The system automatically chooses the velocities at the via points by applying a suita

16、ble 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 points22Higher ord

17、er 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:227.2.3 Higher-order polynomials高階多項式插值高階多項式插值5544332210)(tatatatataat(7.10)7.2 Joi

18、ntSpace Schemes23Where the constraints are given as:237.2.3 Higher-order polynomials(7.11)35243220453423211055443322100020126225432ffffffffffffffftatataaatatatataaatatatatataaa 7.2 JointSpace Schemes24Solution to these equations:247.2.3 Higher-order polynomials(7.12)00100220003320004420005522020(812

19、)(3)23030(1416)(32)21212(66)()2ffffffffffffffffffaaattatttatttat7.2 JointSpace Schemes25Linear 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.257.2.4 Linear function with parabolic

20、blends 用拋物線過渡的線性插值用拋物線過渡的線性插值7.2 JointSpace SchemesDiscontinuous velocity - can not be controlled!26To 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 port

21、ion to change velocity smoothly.267.2.4 Linear function with parabolic blends7.2 JointSpace Schemes27Assume 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

22、about the halfway point.277.2.4 Linear function with parabolic blends7.2 JointSpace Schemes28The velocity at the end of the blend region must equal the velocity of the linear section:287.2.4 Linear function with parabolic blendsbhbbthbttt(7.13)7.2 JointSpace Schemes29Let t=2th ，combining (7.13) and

23、(7.14)297.2.4 Linear function with parabolic blends0)(02fbbttt (7.15) 2)(42022fbttt(7.16)20)(4tf (7.17)The acceleration chosen must be sufficiently high, to ensure the existence of a solution:7.2 JointSpace Schemes30Below shows a set of joint space via points for some joints. Linear functions connec

24、t the via points, and parabolic blend regions are added around each via point.307.2.5 Linear function with parabolic blendsfor a path with via points過路徑點的用拋物線過渡的線性插值過路徑點的用拋物線過渡的線性插值7.2 JointSpace SchemesMulti-segment linear path with blends.31Given:positionsdesired time durations the magnitudes of t

25、he accelerationsCompute:blends timesstraight segment times slopes (velocities)signed accelerations317.2 JointSpace Schemes7.2.5 Linear function with parabolic blendsfor a path with via points32Inside segment:327.2 JointSpace Schemes7.2.5 Linear function with parabolic blendsfor a path with via point

26、skjdjkjkkjkklkkjkklkdjkjkjktttttt2121)sgn( 33First segment:337.2 JointSpace Schemes7.2.5 Linear function with parabolic blendsfor a path with via points211212112121211221212111212121)(2)sgn(tttttttttdddd 34Last segment:347.2 JointSpace Schemes7.2.5 Linear function with parabolic blendsfor a path wit

27、h via points1)1()1()1(1)1(112)1()1(12121)(2)sgn(nnnndnnnnndnnnnnnnnndnndnnnnnttttttttt 35To go through the actual via points:Introduce “Pseudo Via Points”Use sufficiently high acceleration357.2 JointSpace Schemes7.2.5 Linear function with parabolic blendsfor a path with via points36Ch.7 Trajectory P

28、lanning of Robots 36Ch.7 Trajectory Planning of Robots37When 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

29、 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 Schemes377.3 Cartesian-Space Schemes38Description of a task7.3 Cartesian-Space Schemes387.3 Cartesian-Space Sch

30、emes39Cartesian straight line motionMove from point Pi to Pi+1 , which described by relative homogenous transformation:7.3 Cartesian-Space Schemes397.3 Cartesian-Space Schemes10001000iziziziziyiyiyiyixixixixiiiiiBpaonpaonpaonPpaon1000100011111111111111111ziziziziyiyiyiyixixixixiiiiiiBpaonpaonpaonPpa

31、on40In 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. 7.3 Ca

32、rtesian-Space Schemes407.3 Cartesian-Space Schemes41Cartesian planning difficulties (1/3):41Initial (A) and Goal (B) Points are reachable, but intermediate points (C) unreachable.7.3 Cartesian-Space Schemes7.3 Cartesian-Space Schemes4242Approaching singularities some joint velocities go to causing d

33、eviation from the path.7.3 Cartesian-Space Schemes7.3 Cartesian-Space SchemesCartesian planning difficulties (2/3):4343Start point (A) and goal point (B) are reachable in different joint space solutions (The middle points are reachable from below.)7.3 Cartesian-Space Schemes7.3 Cartesian-Space Schem

34、esCartesian planning difficulties (3/3):44Ch.7 Trajectory Planning of Robots 44Ch.7 Trajectory Planning of Robots457.4 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. Th

35、is path generator computes the trajectory at the path update rate.7.4 Path Generation at Run Time45, ,and 467.4.1 Generation of joint space pathsIn 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 Time(7.4)46332210)(tatataattaattataat32232162)(32)( (7.3)47In the case of linear splines with parabolic blends, the value of time, t, is checked on each update to determine whether we are curr

37、ently in the linear or the blend portion of the segment.In the linear portion, the trajectory for each joint is calculated as7.4 Path Generation at Run Time477.4.1 Generation of joint space paths0 jkjkjt(7.41)48In the case of linear splines with parabolic blends, the value of time, t, is checked on

38、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 as7.4 Path Generation at Run Time487.4.1 Generation of joint space paths(7.42)kinbkjkinbkinbjkjtttt 221)(1where2inbjjktttt 49In 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 as7.4 Path Generation at Run Time49(7.44)7