DIEF, Department of Engineering Enzo Ferrari University of Modena e Reggio Emilia Italy Online Trajectory Planning for robotic systems

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1 DIEF, Department of Engineering Enzo Ferrari University of Modena e Reggio Emilia Italy Online Trajectory Planning for robotic systems Luigi Biagiotti Luigi Biagiotti

2 Introduction to online planners Characteristics of online planners Optimal: minimum time trajectories Constraints must be considered Bounded velocities, accelerations, jerks, The user can assign initial i i an final interpolating i conditions i Positions, velocities, accelerations, Computationally efficient Classification Multidimensional planners Scalar planners p(u) Path u(t) Multidimensionality handled with the path-velocity decomposition 2

3 Scalar problem goal: planning online time-optimal trajectories compliant with prescribed constraints on the first n derivatives (n-th order trajectory): Multi-segment trajectories: es. Double S velocity trajectory asymmetric constraints The final position is specified by means of a rough reference input 3

4 Closed-loop and open-loop planners Closed-loop structure Open-loop structure 4

5 Closed-loop generator Modular structure, based on the iteration of the following scheme ( ) and on variable structure controller L. Biagiotti and R. Zanasi, Time-optimal regulation of a chain of integrators with saturated ated input and internal variables: ab An application to trajectory planning, in NOLCOS 2010, Bologna, Italia, 1-3 September

6 Closed-loop generator: Third order filter Bou nd on x2 reac ched Bou nd on x2 not reach ed 6

7 Closed-loop generator: Third-order filter The expression of the switching surface for the third-order filter with the switching surface satisfies the constraint (and obviously ) and forces the state variable to reach the value (and the des Too complex for higher-order trajectory generators 7

8 Open-loop trajectory generator Additional condition symmetric ti constraints t where is the impulse response of the filter with the free parameters that depend on the temporal constraints 8

9 Open-loop trajectory generator In order to guarantee that the trajectory is time-optimal, the following constraints must hold true restriction to the feasible trajectories L. Biagiotti and C. Melchiorri, Fir filters for online trajectory planning pa with time- and frequency-domain specifications, cat s, Control o Engineering Practice,

10 Open-loop trajectory generator Evaluation of the trajectory by means of Finite Impulse Response (FIR) filters: discretization i i by backward-differences method Note that with Moving average filter 10

11 Open-loop trajectory generator Computational complexity: The i-th filter is characterized by the differences equation Note that therefore the evaluation of a trajectory of order n requires n multiplications 2n additions 11

12 Open-loop trajectory generator: Applicative examples Multi-point optimal trajectories Consider a third order trajectory Where are the via-points and Not constant 12

13 A comparison between c.l. and o.l. filters If the hypotheses mentioned in previous slides are met c.l. o.l. ol o.l. chattering 13

14 A comparison between c.l. and o.l. filters If the hypotheses mentioned in previous slides are not met: the input changes before the previous tracts has finished c.l. o.l. Bounds overcome 14

15 A comparison between c.l. and o.l. filters Closed-loop generator: asymmetric constraints and variation of the bounds c.l. c.l. Asymmetric contraints Bound variation 15

16 A comparison between c.l. and o.l. filters Ramp input c.l. o.l. o.l. Tracking error null Tracking error constant 16

17 A comparison between c.l. and o.l. filters Generic input c.l. o.l. Velocity bounded Low-pass filter 17

18 Open-loop trajectory generator Frequency characterization of the trajectory/filter: The Fourier transform of is where By properly choosing Ti it is possible to shape the spectrum of the output trajectory 18

19 Open-loop trajectory generator: Applicative examples Time- and frequency- specifications Given a motion system with an elastic transmission, by properly choosing the parameters Ti it is possible to nullify the (acceleration) spectrum of the trajectory at the resonant frequency 19

20 From Filters for optimal trajectories to B-spline filters 20

21 B-splines basics B-splines of degree p are defined as where are the control points, and the basis functions of degree p: with knots 21

22 B-splines basics B-splines of degree p are defined as where are the control points, and the basis functions of degree p: with knots 22

23 B-splines basics B-splines of degree p are defined as where are the control points, and the basis functions of degree p. Geometrical interpretation 23

24 B-splines basics B-splines of degree p are defined as where are the control points, and the basis functions of degree p. Main (positive) features: Equivalence to splines in polynomial form Independance of the algorithms for control points computation from the spline degree p Possibility of generating local changes by modifying some control points Main drawback: Ealationof Evaluation of the function for a given value of t 24

25 Uniform B-splines Particular case of B-splines, defined for an equally-spaced distribution of the knots, i.e. For uniform B-splines, the basis function of degree p can be defined as 25

26 From uniform basis functions to dynamic filters General case Laplace transform p filters 26

27 Open-loop filter for B-splines generation Particular case of B-splines, defined for an equally-spaced distribution of the knots, i.e. They are equivalent to the output of a chain of filters fed by the sequence of control points p filters In an analytical aytca form (Laplace (apacedomain) 27

28 Selection of the control points Given a set of via-points Goal: to design Interpolation a geometrich tihpath from a set of given points Approximation 28

29 Selection of control points: interpolation The interpolation of is achieved by imposing where are the time instants at which the spline crosses. If p is odd, we assume ; accordingly the values of are 29

30 Selection of control points For p=3 (cubic B-splines) the equations for the control points computation are and lead to the following system The first and the last control points are equal to the related via- points 30

31 General structure of trajectory planner The trajectory planner is composed by an algorithm that transforms the desired points in the set of control points ; a cascade of p moving average filters, where p is the desired degree of the B-spline (the resulting trajectory will be ). Between them, the Sequencer produces the piecewise constant function arranging in a line the control points, each one for a duration of T seconds. The continuity of the trajectory and its derivatives at the start/end point is guaranteed by the filters chain. A separated filters chain is needed for each trajectory component. 31

32 Numerical examples An industrial robot performing a welding operation 32

33 Numerical examples An industrial robot performing a welding operation (quintic B- spline, i.e. p=5) Via-points B-spline Control points Control polygon 33

34 Numerical examples An industrial robot performing a welding operation (quintic B- spline, i.e. p=5) Via-points B-spline Control points Control polygon 34

35 Numerical examples An industrial robot performing a welding operation (quintic B- spline, i.e. p=5) By changing (and therefore ) it is possible to modify the derivatives of the trajectory (i.e. velocity, acceleration, jerk, etc.) according to 35

36 Numerical examples A mobile robot in an environment with obstacles (p=2) Via-points B-spline Control points Control polygon 36

37 Numerical examples A mobile robot in an environment with obstacles (p=2) Via-points B-spline Control points Control polygon Modified control points Local modification of the trajectory 37

38 Choice of the time T In the proposed trajectory planner, T is the unique free parameter (which determines the duration of the trajectory) In the literature the time-distance between knots (usually not constant) is determined with the purpose of: minimizing the trajectory duration and guarantying the compliance with temporal constraints on velocity/acceleration/jerk Proposed approach: to determine T, by taking into account frequency constraints (such as eigenfrequencies of the robotic system) L. Biagiotti and C. Melchiorri, Input shaping via b-spline filters for 3-d trajectory planning, in IEEE/RSJ International Conference on Intelligent Robots and Systems (IROS), San Francisco, California, September

39 Frequency analysis of B-spline trajectories From the analytical expression of the B-spline trajectory the spectrum (Fourier transform) immediately descends where In particular, the magnitude of the frequency response is 39

40 Frequency analysis of B-spline trajectories From the analytical expression of the B-spline trajectory the spectrum (Fourier transform) immediately descends where 40

41 Frequency analysis of B-spline trajectories Idea: to choose T in order to nullify the spectrum of the trajectory at the resonant frequency of vibratory systems This approach is very similar to input shaping technique, consisting in filtering i the reference input with a filter, that t depends on the system, in order to avoid residual vibrations 41

42 Filtering characteristics of B-spline generator When applied to an undamped second order system, with natural frequency, the Percent Residual Vibrations (PVR) is no residual vibration for low level l of vibration for 42

43 Comparison with standard Input Shapers PRV of B-spline cubic filter and standard Input Shapers, i.e. P = 1 P = 2 P = 3 P = 4 Strong asymmetry of cubic B-spline PRV characteristics Robustness w.r.t. errors (or changes) on estimation 43

44 Filtering characteristics of B-spline generator Robustness w.r.t. errors on estimation 44

45 Application to a robot with elastic joints Given the general model of a robot with elastic joints Hypotheses: Cartesian structure viscoelastic coupling An ideal control system is able to impose to the actuators the desired trajectory, that is robot dynamics motors dynamics 45

46 Application to a robot with elastic joints Given the general model of a robot with elastic joints Hypotheses: Cartesian structure An ideal control system is able to impose to the actuators the desired trajectory, that is The model of robot + control becomes with 46

47 Application to a robot with elastic joints The robot behaves like a second-order system with The reference path is 47

48 Application to a robot with elastic joints Residual vibration with B-splines along x axis 48

49 Application to a robot with elastic joints Residual vibration with a cubic B-spline along x and y directions limited vibrations if Residual vibrations as a function of the degree p 49

50 Application to a robot with elastic joints B-splines vs. Input shapers: path tracking Cubic B-spline Quintic B-spline ZV IS ZVDD IS No interpolation of the via-points 50

51 Conclusions Open-loop and closed loop enjoy peculiar features that make each type of generator preferable for a specific application: open-loop systems have a very simple structure and low computational costs but they suffers from some limitations (symmetric constraints on velocity, acceleration, etc.) closed-loop trajectory generators allow to modify the limits in runtime and to start a new motion at any time. Open-loop filters may be preferable in all those applications that are affected by vibrations and resonances, since the linear structure allows a precise frequency characterization of the output trajectory Filters for uniform B-splines generation, that are characterized by a similar structure, can be designed with the purpose of properly shaping their frequency spectrum and suppressing residual vibrations, that may be present because elastic phenomena affecting the robotic system. 51

52 Spline trajectories pp-form Multipoint trajectories obtained by joining n-1 polynomial functions (of typical degree 3 or 5) that interpolate n given points and are continuous in the interior i (up to a given order). Splines are the interplating curves with minimum curvature, for a given levell of continuity. it 52

53 Spline trajectories pp-form In order to guarantee the continuity of velocity and acceleration, polynomial of degree 3 are usually considered (cubic splines). Cubic splines are defined as It is necessary to compute 4 coefficients for each polynomial. Therefore there are 4(n-1) parameters to be defined The constraints are : interpolating conditions continuity conditions on the velocity in the internal pointsintermedi continuity conditions on the accleration in the internal points There are degrees of freedom that can be used to set e.g. initial and final velocities Back to B-splines 53

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