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 pathvelocity decomposition 2
3 Scalar problem goal: planning online timeoptimal trajectories compliant with prescribed constraints on the first n derivatives (nth order trajectory): Multisegment trajectories: es. Double S velocity trajectory asymmetric constraints The final position is specified by means of a rough reference input 3
4 Closedloop and openloop planners Closedloop structure Openloop structure 4
5 Closedloop generator Modular structure, based on the iteration of the following scheme ( ) and on variable structure controller L. Biagiotti and R. Zanasi, Timeoptimal regulation of a chain of integrators with saturated ated input and internal variables: ab An application to trajectory planning, in NOLCOS 2010, Bologna, Italia, 13 September
6 Closedloop generator: Third order filter Bou nd on x2 reac ched Bou nd on x2 not reach ed 6
7 Closedloop generator: Thirdorder filter The expression of the switching surface for the thirdorder 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 higherorder trajectory generators 7
8 Openloop 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 Openloop trajectory generator In order to guarantee that the trajectory is timeoptimal, 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 frequencydomain specifications, cat s, Control o Engineering Practice,
10 Openloop trajectory generator Evaluation of the trajectory by means of Finite Impulse Response (FIR) filters: discretization i i by backwarddifferences method Note that with Moving average filter 10
11 Openloop trajectory generator Computational complexity: The ith 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 Openloop trajectory generator: Applicative examples Multipoint optimal trajectories Consider a third order trajectory Where are the viapoints 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 Closedloop 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 Lowpass filter 17
18 Openloop 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 Openloop 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 Bspline filters 20
21 Bsplines basics Bsplines of degree p are defined as where are the control points, and the basis functions of degree p: with knots 21
22 Bsplines basics Bsplines of degree p are defined as where are the control points, and the basis functions of degree p: with knots 22
23 Bsplines basics Bsplines of degree p are defined as where are the control points, and the basis functions of degree p. Geometrical interpretation 23
24 Bsplines basics Bsplines 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 Bsplines Particular case of Bsplines, defined for an equallyspaced distribution of the knots, i.e. For uniform Bsplines, 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 Openloop filter for Bsplines generation Particular case of Bsplines, defined for an equallyspaced 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 viapoints 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 Bsplines) 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 Bspline (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) Viapoints Bspline Control points Control polygon 33
34 Numerical examples An industrial robot performing a welding operation (quintic B spline, i.e. p=5) Viapoints Bspline 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) Viapoints Bspline Control points Control polygon 36
37 Numerical examples A mobile robot in an environment with obstacles (p=2) Viapoints Bspline 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 timedistance 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 bspline filters for 3d trajectory planning, in IEEE/RSJ International Conference on Intelligent Robots and Systems (IROS), San Francisco, California, September
39 Frequency analysis of Bspline trajectories From the analytical expression of the Bspline trajectory the spectrum (Fourier transform) immediately descends where In particular, the magnitude of the frequency response is 39
40 Frequency analysis of Bspline trajectories From the analytical expression of the Bspline trajectory the spectrum (Fourier transform) immediately descends where 40
41 Frequency analysis of Bspline 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 Bspline 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 Bspline cubic filter and standard Input Shapers, i.e. P = 1 P = 2 P = 3 P = 4 Strong asymmetry of cubic Bspline PRV characteristics Robustness w.r.t. errors (or changes) on estimation 43
44 Filtering characteristics of Bspline 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 secondorder system with The reference path is 47
48 Application to a robot with elastic joints Residual vibration with Bsplines along x axis 48
49 Application to a robot with elastic joints Residual vibration with a cubic Bspline 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 Bsplines vs. Input shapers: path tracking Cubic Bspline Quintic Bspline ZV IS ZVDD IS No interpolation of the viapoints 50
51 Conclusions Openloop and closed loop enjoy peculiar features that make each type of generator preferable for a specific application: openloop systems have a very simple structure and low computational costs but they suffers from some limitations (symmetric constraints on velocity, acceleration, etc.) closedloop trajectory generators allow to modify the limits in runtime and to start a new motion at any time. Openloop 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 Bsplines 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 ppform Multipoint trajectories obtained by joining n1 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 ppform 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(n1) 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 Bsplines 53
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