Mapping Accuracy of Trajectories of Manipulator Motion

The paper presents the algorithm for trajectory planning and the analysis of motion accuracy of an anthropomorphic manipulator whose the end-effector moves over the rectilinear-arc path. The path consists of two rectilinear segments of intersecting directions coupled with the arc of a fixed radius. On these rectilinear segments, the end-effector acceleration is described by a polynomial of 7-degree. The motion over the arc proceeds at a constant velocity. The dynamic analysis was carried out on the basis of Lagrange`s equations of the second order. Manipulator flexibility and damping were considered according to the Kelvin-Voigt model introducing spring-damping components into the drive system. The simulation test results were presented in the form of spatial courses of the pre-assigned and realized trajectories as well as time courses of trajectory end-effector mapping errors for three paths with different radii of the arc.


Introduction
One of the fundamental tasks in trajectory planning operation proves to be the analysis of errors resulting from the manipulator dynamic chain deformability.Manipulator structure, geometry, mass distribution as well as its shape and position in the task space of the desired path have, besides the stiffness and damping, a significant effect on the mappability of the prescribed trajectory.Erkorkmaz et al. [5] proposed a trajectory planning strategy for maintaining the tool positioning accuracy in high speed cornering applications.A 3D contour error estimation algorithm was presented for determining the geometric deviation from the arbitrarily shaped toolpaths.Two spline fitting strategies were developed for smoothening sharp corners.Dong et al. [4] examined the possibility of scheduling or varying the feed-rate by taking into consideration the geometry of the contour that the machine is expected to follow and the physical capabilities of the machine.The paper presents, the constraints on permissible jerk and as a consequence, machine vibrations got reduced, element wear declined and mapping errors in the end-effector path limited.Tseng et al. [8] suggested an algorithm for the jerk-limited acceleration planning.With this algorithm, both the chord error, the maximum acceleration and jerk were within the permissible limits.
Apart from the factors mentioned at the beginning, which influence the accuracy of movement, the way of trajectory parameterization is also very important.Among the applied methods, parameterization with the use of higher degree polynomials is worthy of mention.Thus way it is possible to obtain smooth trajectories with simultaneous limitation of speed, acceleration or jerk.Boryga and Graboś [1] used polynomials of higher degree for trajectory planning of robot manipulator.The linear acceleration profiles of end-effector were planned as the polynomials of 9, 7 and 5 degree.They also demonstrated the algorithm for Polynomial Cross Method (PCM) to plan the end-effector motion whose path is composed of two rectilinear segments [6].Boryga et al. [2] applied the Planning Rectilinear-Arc Polynomial Trajectory (PR-APT) method for planning the end-effector motion trajectory during harvesting tomatoes in the greenhouse.The acceleration profile of end-effector was planned as the polynomial of 7 th -degree.Milica et al. [7] presented the method for determining the optimal trajectory of a parallel robot.The Hermite polynomial expressions were used as the time function of geometrical parameters.The authors concluded that the use of polynomial functions with the simplest expressions leads to minimization of effort associated with the analysis and mathematical formulation of the optimization problem.Chen et al. [3] proposed the Intermediate Point Obstacle Avoidance (IPOA) algorithm to provide smooth trajectory and efficient movement while avoiding obstacles.Combining the IPOA algorithm in a series, multi-obstacle avoidance by the robotic arm in the complex environment can be accomplished.
The paper is arranged as follows.Chapter II proposes an innovative and very effective algorithm for planning rectilinear-arc trajectories of manipulator end-effector.On the rectilinear segments the acceleration course is described by the polynomial of 7 th -degree.Transition from the rectilinear segment to the arc and vice versa proceeds without a change of velocity.Chapter III presents the dynamics model.Flexibility and damping forces in kinematic pairs were taken into consideration introducing spring-damping components into the drive system.Chapter IV depicts the numerical example.For the planned rectilinear-arc trajectory (for three arc radii), prescribed and realized trajectories of end-effector and time courses of position errors were presented.

Planning of rectilinear-arc trajectories of end-effector motion 1.1 Assumptions
The first stage in planning a rectilinear-arc trajectory is as follows:  Coordinates of the initial B, final E and intermediate M points.The points B and E should be located in the workspace of the robot manipulator.The point M is the intersection point of the rectilinear segments and does not lie on the planned path.It can be situated outside the workspace (Fig. 1). Arc radius R.  Maximum acceleration on the rectilinear segments amax.

Algorithm for trajectory planning
Step 1. Determine the scalar equation of the plane  passing through the points B, M, and E.
Step 2. Define the parametric equations of the straight line passing through the points B and M as well as points E and M. Step 5. Define the coordinates of the center of the arc O that lies in the plane  and is tangential to the points T1 and T2 to the segments BM and ME.
Step 6. Determine the apex BME and the apex T1OT2 angles.
Step 7. Determine matrix transformation A between the local system of coordinates ξ1ξ2ξ3 connected with arc and the reference coordinate system x1x2x3.The origin of the system ξ1ξ2ξ3 is the point O, versor of ξ1 axis oriented towards the point M and the direction of ξ2 axis versor intersecting the line passing through the points B and M. The versor of ξ3 axis is a complement of the right-handed coordinate system.
Step 8. Determine path increments on the rectilinear segments s BT1 and s T2E .
Step 9. Pre-establish motion time for the BT1 and T2E segments For the segment BT1: t=t BT1 , s=s BT1 .For the segment T2E: t=t T2E , s=s T2E .
Step 11.Determine motion time on the BT1 and T2E segment.If vT1vT2, then velocity in the tangency points and over the arc should be assumed to be v Arc =vT2 and the motion time on the BT1 segment is derived from the dependency (2) where t=t BT1 , s=s BT1 and v=vT2 should be substituted.In the case vT1<vT2, the velocity at the tangential points and on the arc should be assumed to be v Arc =vT1 while the motion time on T2E segment is derived from the dependency (2) where t=t T2E , s=s T2E and v=vT1 should be substituted.Step 12. Establish the polynomial coefficient for calculation of acceleration, speed and position for the BT1 and T2E segments 9 315 8 For the segment BT1: p=p BT1 , s=s BT1 and t=t BT1 .For the segment T2E: p=p T2E , s=s T2E and t=t T2E .
Step 13.Determine the motion duration time over the arc Step 14. Define the dependence on angular displacement in motion over the arc Step 15.Determine the coordinates of the vectors of position  Arc , speed ' Arc and acceleration '' Arc of the motion over the arc in the coordinate system where: Step 16.Determine translation in time on the T2E segment Step 17. Perform translation in time of the polynomial depicting the acceleration profile for the T2E segment by tb value Step 18. Set the total motion time on the designated trajectory

Dynamics of the open kinematic chain
The motion equations were derived on the basis of the Lagrange equations of the second order.The general form of the equations is written as follows where: B(θ) -the inertia matrix, O(θ) -the matrix of centrifugal torques, C(θ) -the matrix of Coriolis torques, G(θ) -the vector of gravity torques, τ -the generalized joint force vector.
4 The dynamics model takes into account flexibility and damping effects introducing deformable objects (Kelvin-Voigt model) between the drive units and links.The forces generalized in i-n th kinematic pair were defined by the equation

Numerical example
The simulation tests were performed for three degrees of freedom anthropomorphic manipulator (Fig. 2).Geometrical and mass values are presented in Table 1.The same value for the stiffness coefficient was assumed for all the links, i.e. ks =2500 Nm/rad and for the viscous damping coefficient cd =250 Nms/rad.The coordinates of B, E and M points are B(0.5,0.5,1),E(0.75,0.75,1),M(0.5,0.75,1).The assumed maximum acceleration was amax =0.25m/s 2 .The simulation tests were conducted for three paths with the arc radii equal to: R1=0.025m (Path 1), R2=0.01m (Path 2) and R3=0.005m (Path 3).The coordinates of the characteristic points of trajectory are presented in Table 2.The polynomial coefficients, motion times and motion velocities over the arcs are presented in Table 3.The maximum absolute mapping errors of trajectory are presented in Table 4.

Conclusions
Based on the analysis using the proposed algorithm for planning of rectilinear-arc trajectories of the end-effector there can be drawn, the following conclusions:  The arc radius was found to have a minor effect on the total motion time and at the same time significant influence on position errors.The change of the arc radius from R1=0.025m (Path 1) to R2=0.01m (Path 2) results in the about 2-fold increase in the trajectory mapping errors toward x1 and x2 while from R1=0.025m (Path 1) to R3=0.005m (Path 3) -an approximately 4-fold increase.Motion time growth was equal to 0.001s and 0.002s, respectively. The errors on the x1 and x2 directions result from dynamic loads as the pre-assigned motion proceeded in the x1x2 plane.The error toward the x3 axis is mainly associated with the static load and proved to be almost the same for each arc radius. Owing to the assumed small linear acceleration, the dynamic load errors (inertia, centrifugal and Coriolis forces) are smaller than those caused by the static load. The errors toward the x3 axis result from the angular position errors of the second and third links whereas those toward the x1 and x2 axes arise chiefly from the angular position error of the first link.

2 ITMFig. 1 .
Fig. 1.The rectilinear-arc motion path with the added coordinate systems Step 3. Establish equations of the 1 and 2 planes perpendicular to the straight lines passing through the points B and M (1 planes) as well as the points E and M (2 planes) and those passing through the center of the arc O.Step 4. Determine the coordinates of point T1 as common to the plane 1 and the straight line running through the points B and M as well as the coordinates of the point T2 as the common to the plane 2 and the straight line passing through the points E and M.Step 5. Define the coordinates of the center of the arc O that lies in the plane  and is

5 ITMTable 2 .
Web of Conferences 21, 00009 (2018) CST 2018 https://doi.org/10.1051/itmconf/20182100009The coordinates of characteristic points of paths Path Tangent point T1 Tangent point T2 Arc center O determine the pre-assigned angular displacements at each kinematic pair g1, g2, g3 the inverse kinematics problem was solved.There were established the accomplished angular displacements r1, r2, r3 as a result of numerical integration of the motion equations (11) and the end-effector positions xr1, xr2, xr3 on the basis of the simple problem of kinematics.The end-effector position errors xi were calculated from the dependence xi=xgi -xri for i = 1, 2, 3.The simulation experiments provided the courses of the pre-assigned and accomplished trajectory of end-effector motion and the time courses of end-effector position errors (Fig.3).

Table 1 .
The geometrical and mass values of manipulator

Table 3 .
The characteristic values of the planned trajectories

Table 4 .
The maximum absolute mapping errors of trajectory