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In fig. 4.3 we present the desired yg versus the 7 variable. Figure 4.3: First Path - Evolution of the desired output versus the path variable — Figure 4 In fig. 4.3 we present the desired yg versus the 7 variable. Figure 4.3: First Path - Evolution of the desired output versus the path variable

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Figure 1.2: The Robodoc surgical system.

Figure 2.1: IST planar flexible link robot manipulator Figure 2.2: Mechanical model of a single flexible link The kinetic energy of the system can deduced as [16]

being k the total number of assumed modes. The mass matrix and the stiffness matrix are given by The inertial displacement at a distance x from the frame origin in the OX direction is given by Here, 7; is the modal amplitude of the ith mode considering the assumed modes discretization procedure

Figure 2.3: The first two mode shapes 2.2 IST planar flexible robot manipulator The state space representation of the flexible robot manipulator can be represented as,

Throughout this work, the above linear model is used for controller design, whereas a non-linear model (quadratic displacements) is used for simulation.

Table 2.1: IST flexible robot manipulator characteristics suffice. In table 2.1 the most important properties of the IST planar manipulator are presented.

End-effector motion planning for a one link non-minimum phase robot Figure 3.1: Location of the zeros and poles for transfer function (3.16) associated to the initial conditions:

Note that these conditions were chosen to force the desired torque to be symmetric. The desired a coefficients are then directly obtained, solving the linear system above. In this way it’s possible to calculate 3.2 Stable inversion of the IST planar flexible link robot manipulator

End-effector motion planning for a one link non-minimum phase robot

Figure 3.2: Desired input 1. A partial state feedback, based on the joint position and velocity variables.

The method was applied for tr = 2.78 and yr = —0.35m (fig. 3.2 and fig. 3.3). The closed loop control

Figure 3.5: Simulated displacement of the End-Effector

End-effector motion planning for a one link non-minimum phase robot Fig. 3.6 presents the output torque of the close-loop form. Indiscernible in this figure, are the smal

Figure 3.7: Simulated output with an external input torque (Amplitude 0.002Nm)

Figure 3.8: Simulated close-loop torque with an external input torque (Amplitude 0.002Nm)

Figure 3.9: Simulation tracking error for an external input torque (Amplitude 0.002Nm) End-effector motion planning for a one link non-minimum phase robot

Figure 3.10: Simulated close-loop torque with an external input torque (Amplitude 0.1Nm)

Figure 3.11: Simulation output with an external input torque (Amplitude 0.1Nm)

Figure 3.12: Simulation tracking error for an external input torque (Amplitude 0.1Nm)

produces an undesired response of the closed-loop torque. Figure 3.13: Simulated close-loop torque with noise

Figure 3.14: Simulation output with noise

Figure 3.16: Simulated close-loop torque with miscalculated zeros

Figure 3.15: Simulation tracking error with noise

Figure 3.17: Simulation output with miscalculated zeros End-effector motion planning for a one link non-minimum phase robot

Figure 3.18: Simulation tracking error with miscalculated zeros

End-effector motion planning for a one link non-minimum phase robot _ The calculation of the proportional part of the controller is obtained by the following transfer function

End-effector motion planning for a one link non-minimum phase robot As in the previous validation, the effects of external factors, such as noise, external torque input, and miss calculation of the transfer function zeros will be analyzed here.

Figure 3.24: Simulated output with an external input torque (Amplitude 0.002Nm) Introducing an input torque starting at t = 1s and ending at t = 2.4s, with an amplitude of 0.002N/m,

Figure 3.25: Simulated close-loop torque with an external input torque (Amplitude 0.002Nm)

Figure 3.26: Simulation tracking error for an external input torque (Amplitude 0.002Nm)

Figure 3.27: Simulated output with an external input torque (Amplitude 0.1Nm) external input was applied (t=1s and t=1.4s).

Figure 3.28: Simulated close-loop torque with an external input torque (Amplitude 0.1Nm)

Figure 3.30: Simulated close-loop torque with noise

Figure 3.29: Simulation tracking error for an external input torque (Amplitude 0.1Nm)

Figure 3.31: Simulation output with noise End-effector motion planning for a one link non-minimum phase robot

Figure 3.32: Simulation tracking error with noise

Figure 3.33: location of zeros and poles in a closed loop form for miss calculation of transfer function The location of the zeros and the poles is represented in fig. 3.33: As before the miscalculation of the transfer function zeros for this case does not pose a significan

Figure 3.34: Simulated closed-loop torque with miscalculated zeros End-effector motion planning for a one link non-minimum phase robot

Figure 3.35: Simulation output with miscalculated zeros

Path-Following for one link non-minimum phase robot one solution to achieve a path controller for (4.1), such that the close loop state is bounded detectable, the number of inputs is as large as the number of outputs (mm > q) and the zeros of (A, B, C, D

Figure 4.2: The path in x, y coordinates the obtained path in x, y coordinates is the arc represented in fig. 4.2. while a2 smoothens the square wave, and the m, variable constrains the maximum speed. As expected For the results presented in the next section, the following data was used. The system state space where the parameter a, sets the width of the low speed regions at the beginning and the end of the path, epresentation in continuous mode is

Figure 4.4: First Path - Simulation tracking error

Figure 4.5: First Path - Deformation of the end-effector

Figure 4.6: First Path - Evolution of the desired path variable versus the real path variable Path-Following for one link non-minimum phase robot Second path simulation results

degree rotation. As it is seen in fig. 4.7, the desired velocity profile has non-zero initial and final conditions. Figure 4.7: Second Path - Simulation velocity profile

value, and it’s cause can be due to the acceleration profile, since it has non zero initial and final values Figure 4.8: Second Path - Simulation tracking error

Path-Following for one link non-minimum phase robot Figure 4.9: Second Path - Evolution of the desired path variable versus the real path variable

Figure 4.10: Second Path - Deformation of the end-effector

Figure 4.11: Second Path - Simulation velocity profile 4.1 Path-Following method with internal model control

Figure 4.12: Second Path - Deformation of the end-effector

Figure 4.13: Second Path - Simulation tracking error

Figure 4.14: First Path validation - Simulation input torque (White noise with the noise power of 0.1) These values can be considered as a reasonable external perturbation. introduced. The external perturbation torque is 5 times greater that the normal input torque (fig. 4.14)

Figure 4.15: First Path validation - Deformation of the end-effector (White noise with the noise power of 4.1 Path-Following method with internal model control

Figure 4.16: First Path validation - Simulation output (White noise with the noise power of 0.1)

Figure 4.17: First Path validation - Simulation tracking error (White noise with the noise power of 0.1) Path-Following for one link non-minimum phase robot

Figure 4.18: Second Path validation - Simulation tracking error (External step input torque with amplitude \ step input torque with the periodicity of 2s and amplitude of 1Nm has been added to the system input

Figure 4.19: Second Path validation - Simulation input torque (External step input torque with amplitude 4.1 Path-Following method with internal model control

Figure 4.20: Second Path validation - Deformation of the end-effector (External step input torque with

Figure 4.21: Second Path validation - Simulation Tracking error (White noise with the noise power of 1)

Figure 4.22: Second Path validation - Simulation input torque (Path properties: a; = 1, ag = 0.1 and external input torque remains the same

Figure 4.23: Second Path validation - Deformation of the end-effector (Path properties: aj = 1, ag = 0.1 values increase, at the same time as the desired theta limitation occurs.

Figure 4.24: Second Path validation - Simulation tracking error (Path properties: aj = 1, ag = 0.1 anc

Path-Following for one link non-minimum phase robot Figure 4.25: Second Path validation - Evolution of the desired path variable versus the real path variable

Figure 4.26: Second Path validation - Simulation output (Path properties: a; = 1, ag = 0.1 and m, = 1

Figure 4.27: Second Path validation - Joint velocity versus error (Path properties: a; = 1, ag = 0.1 anc Normal Form Path Following

which controls the leader’s motion in the following system The trajectories of system (4.27) - (4.29) are related to the leader's motion via the error coordinates MIMO and SISO systems to strictly proper systems (Normal form representation). lind this representation can be found in [19]. There, it is explained the structural decomposition of

functions, the M matrix is assumed with the following structure torque vector. Here we have included a damping matrix C. Furthermore to contemplate any type of shape

For n sample times during trajectory execution, data for g, g , ¢ and 7 must be acquired. Renaming Transforming the system equations into state space representation we obtain and the T vector represents the system input

Simplifying equation (4.49) yields M;'= = Gi;, from equation (4.45) and (4.48) the following equation is obtained:

Path-Following for one link non-minimum phase robot therefore using a Recursive Least Square (RLS) method (Appendix B), where all the data acquired during the time of trajectory execution was used, we may obtain the following variables.

Figure 4.28: Output values of flexible link identification

Figure 4.30: Flexible link identification error

Figure 4.29: Flexible link identification validation

To identify the system, an ARMAX model with a time sample of 0.001, and a Tustin transformatior The identification of the system was obtained using an input step with amplitude of 0,02rad and a period from discrete to continuos model was used. This model had the following properties. The system transfer function considered had a 6 order denominator, a 5 order numerator, considering a noise transfer function with a 6 order denominator and a 4 order numerator and with one time sample delay

Figure 5.3: Location of zeros and poles in a close loop form Fig. 5.3 presents zeros and poles of the system.

Figure 5.5: End-effector vibration at high average velocity

Figure 5.4: End-effector vibration at low average velocity

Figure 5.6: Input torque at low average velocity

Figure 5.7: Input torque at high average velocity

[he new state space, input and output of the decomposed system are described by the following set of

Where (Aaa) contains all the system invariant zeros and nq is the relative degree of T Figure A.1: Interpretation of structural decomposition of a SISO system

is nonsingular. Zo can be chosen where whose rows form a basis of the null space of Za. Being 2. Construction of a preliminary state transformation Let Z) be an ng X n constant matrix, where n,

algorithm: problem of inverting the matrix at each sample time that reduces the computational efficiency of the Equation (B.14) gives a direct update from B(t — 1) to B(t), but the update of P(t — 1) emerge the