Tesi dell’Ing. Jason Bettega presentazione del 20 aprile 2024 al Collegio Ingegneri Venezia

DOCTORAL COURSE IN MECHATRONICS

AND PRODUCT INNOVATION ENGINEERING

Model-based methods for precise trajectory tracking in underactuated multibody systems

Ph.D. Candidate: Ing. Jason Bettega

Supervisor: Prof. Dario Richiedei

1. Introduction and definition of underactuated mechanical systems

2. Feedback control

• Proposed Model Predictive Control formulation

• Test cases

3. Feedforward control

• Proposed inverse dynamics formulation

• Test cases

4. Conclusions

1. Introduction and definition of underactuated mechanical systems

2. Feedback control

• Proposed Model Predictive Control formulation

• Test cases

3. Feedforward control

• Proposed inverse dynamics formulation

• Test cases

4. Conclusions

Underactuated mechanical systems characterize different engineering areas; indeed, several examples can be taken from real world applications.

Pickandplacetasks

➢ Underactuated mechanical systems

Underactuated multibody systems are characterized by having less actuators than degrees of freedom.

• Intentionally, by a mass or cost driven reduction of actuators

• Accidentally, by the failure of one or several actuators

• Structurally, due to the flexibility of light-weight components

=

➢ Underactuated mechanical systems

Due to the underactuation property, this kind of systems can be characterized by unstable internal dynamics, which consists in the presence of Right Half Plane (RHP) zeros in the linearized counterpart, instead of Left Half Plane (LHP) ones.

Example:

Systems characterized by the presence of RHP zeros are affected by 3 main issues:

• Model inversion cannot be done exactly

• Delay in tracking responses

• Presence of undershoot

The control of underactuated mechanical systems still represents an open problem in the research field.

Feedbackandfeedforwardcontrolmustbeinvestigated.

1. Introduction and definition of underactuated mechanical systems

2. Feedback control

• Proposed Model Predictive Control formulation

• Test cases

3. Feedforward control

• Proposed inverse dynamics formulation

• Test cases

4. Conclusions

➢ Introduction to Model Predictive Control

Among the several common controllers (LQR, Pole Placement, …), Model Predictive Control has been privileged due to two main reasons:

• its ability to achieve an optimal control input at each time step

• its capability to include constraints into the optimization problem

In particular, with the goal to achieve good performances in the presence of time-varying trajectory tracking tasks, a novel MPC formulation has been proposed, which can be represented by the following general block scheme:

Autonomous system + Underactuated system

Predictive Control

➢ Introduction to Model Predictive Control

The following 2-dof underactuated robotic arm is considered hereinafter, although the method is general. �� = active joint �� = passive joint

τ = motor torque

1 = actuated angle

2 = non-actuated angle

• Step 1: dynamic model

Goal: perform a pick-and-place task from a starting position (pick location) to an ending one (place location), repetitively, ensuring low tracking error. (second-order dynamic model) (state-space model)

➢ Introduction to Model Predictive Control

• Step 2: cost function

= desired output

= input weigth

The standard MPC formulation defines the desired trajectory along the prediction horizon as follows:

Butisthisdefinitionthe rightone?

Introduction to Model Predictive Control

Through the implementation of the standard MPC formulation, the following system response is achieved:

Tracking delay

Good performances in terms of path tracking

Howtosolve thisproblem?

Bad performances in terms of trajectory tracking

• Presence of right half plane zeros (non-minimum phase system)

• Wrong definition of the desired output along the prediction horizon

➢ MPC-ERD formulation

In order to overcome this issue, a novel MPC formulation (Model Predictive Control with Embedded Reference Dynamics) is proposed where the desired output along the prediction horizon is redefined through an autonomous state-space model:

where ���� is the matrix responsible for the dynamics of the reference trajectory. The evaluation of the matrix ���� is carried out through a technique known as «Dynamic Mode Decomposition», which firstly appeared in the fluid dynamics community.

Whynotapplying theseconceptsin motioncontrol?

MPC-ERD formulation

Once ���� is evaluated and ���� is chosen to make the couple (����,����) observable, the predicted reference is reformulated as follows:

where:

The huge improvement is firstly verified by comparing the predicted reference in both formulations:

Howdoestheclosedloopsystembehave?

test case: experimental results

In order to fully evaluate the proposed controller, both MPC algorithms have been implemented to the real setup available in the laboratory:

test case: experimental results

The real tracking responses are reported in the followings, and it can be observed that the supremacy of the proposed controller has been verified.

test case: experimental results

The real tracking errors are also reported for sake of completeness, showing an approximative improvement of one order of magnitude.

test case: experimental results

To have a better comprehension of the improvement achieved with the proposed control algorithm, the relative tracking responses in the Cartesian plane are here reported.

➢

2nd test case: Cable Suspended Parallel Robot

To further validate the proposed control algorithm, a Cable Suspended Parallel Robot is considered, which represents a very challenging test case:

• Cable tensions greater than zero

• Proper trajectory planning (admissible workspace)

• Proper controller design (lack of positive controllability)

End-effector

= tip position, speed and acceleration, respectively

Since the dynamic of these systems is highly nonlinear, a 2-stage controller strategy is proposed. ��

= optimal cable tensions

= motor position and speed, respectively

= computed motor torques

➢ 2nd test case: MPC-ERD with

CSPR

• First stage: MPC-ERD

MPC with Embedded Reference Dynamics and Embedded Integrator is adopted, with two proper modifications:

1) time-varying state-space formulation to handle nonlinearities

2) DMD is applied to each cartesian axis, to avoid rank deficient matrices, and finally ���� is achieved

• Second stage: Motor dynamics model

Once the optimal cable tensions are computed by the MPC-ERD , the commanded motor torques are computed through an inverse-dynamic approach considering the dynamic model of each actuator:

The desired accelerations

obtained through inverse kinematics of the load reference trajectory.

Howarethetrajectorytrackingperformancesofthiscontroller?

2nd test case: numerical results

Both standard MPC formulation with embedded integrator and MPC-ERD have been tested in this scenario and their tracking results are here reported.

➢ 2nd test case: numerical results

In particular, these results have been achieved by embedding the motor characteristic curve as an additional constraint.

By looking at the cable tensions it can be noticed that MPCERD is able to achieve good trajectory tracking performances while ensuring proper safety conditions.

➢ MPC-ERD: key aspects

Summarizing, thanks to the proposed MPC-ERD algorithm:

✓ Dynamic model of the underactuated system is embedded into the controller design

✓ Input constraints are embedded into the controller design, achieving an optimal constrained solution

✓ Reference dynamics are embedded into MPC-ERD, allowing for a better control evaluation during the prediction horizon

Increased safety level

1. Introduction and definition of underactuated mechanical systems

2. Feedback control

• Proposed Model Predictive Control formulation

• Test cases

3. Feedforward control

• Proposed inverse dynamics formulation

• Test cases

4. Conclusions

➢ Feedforward formulation

The control of underactuated systems usually lead to the following challenges: Control of non minimum phase systems Feedforward computation through inverse dynamics

To solve inverse dynamics, a feedforward method has been proposed, which consists of six steps:

1) Partitioning of the nonlinear dynamic model into actuated and unactuated coordinates (through QRdecomposition)

2) Linearization of the desired output �� and exploitation of the linearly combined output technique

3) Evaluation of the internal dynamics

4) Selection of the parameter �� to make the internal dynamics stable and evaluation of

through a differential approach

5) Evaluation of ��

,���� thanks to the nonlinear kinematic equations

6) Evaluation of the feedforward input forces through an algebraic approach

Istheproposedtechniqueeffective?

experimental results

The enhanced feedforward technique has been tested both numerically and experimentally on a 2-dof underactuated robotic arm with the goal to perform a pick-and-place task, considering different opening angles and including steady intervals. The experimental results confirm the correctness of the model inversion.

Andiftherequiredcontrolforcescannotbedirectlyapplied?

➢ Trajectory planning

Usually, industrial robots are designed in such a way that the control torques/forces cannot be directly modified.

To overcome this issue, the differential part of the proposed inverse dynamics method can be used for trajectory planning purposes.

test case: experimental results

The proposed trajectory planner has been experimentally validated considering a pendulum moved by a parallel robot (Adept Quattro) which emulates an overhead cartesian crane.

test case: experimental results

The proposed trajectory planner has been experimentally validated considering a pendulum moved by a parallel robot (Adept Quattro) which emulates an overhead cartesian crane. Considering underactuation (proposedmethod) Neglecting underactuation

1. Introduction and definition of underactuated mechanical systems

2. Feedback control

• Proposed Model Predictive Control formulation

• Test cases

3. Feedforward control

• Proposed inverse dynamics formulation

• Test cases

4. Conclusions

Conclusions

✓ A general frame for motion control of underactuated systems has been developed by focusing the attention on:

Feedback Control (MPC-ERD)

Feedforward Control

(Inverse dynamics)

✓ Additionally, the attention has focused on fully-actuated systems, such as Cable Driven Robots, due to their challenging nature, that could lead to the presence of underactuated systems if cable tensions would become negative.

1) Bettega J., Richiedei D., Trevisani A., Path tracking in cable suspended parallel robots through position-dependent model predictive control with embedded integrator, Proceedings of the 10th Eccomas Thematic Conference on Multibody Dynamics(2021)

2) Araújo J.M., Bettega J., Dantas N.J.B., Dórea C.E.T., Richiedei D., Tamellin I., Vibration control of a two-link flexible robot arm with time delay through the robust receptance method, Applied Sciences (2021)

3) Bettega J., Richiedei D., Trevisani A., Using pose-dependent model predictive control for path tracking with bounded tensions in a 3DOF spatial cable suspended parallel robot, Machines (2022)

4) Bettega J., Richiedei D., Trevisani A., Feedforward control of a nonlinear underactuated multibody system, Recent Trends in Wave Mechanics and Vibrations. WMVC 2022. Mechanisms and Machine Science (2023)

5) Bettega J., Richiedei D., Tamellin I., Trevisani A., Motion Planning Through Model Inversion for a Gantry Crane Moving a Double Pendulum. In: Petrič, T., Ude, A., Žlajpah, L. (eds) Advances in Service and Industrial Robotics. RAAD 2023. (2023)

6) Bettega J., Richiedei D., Trajectory tracking in an underactuated, non-minimum phase two-link multibody system through model predictive control with embedded reference dynamics, Mechanism and Machine Theory (2023)

7) Bettega J., Boschetti G., Piva G., Richiedei D., Trevisani A., Reconfiguration strategy for fully actuated translational cable-suspended parallel robots, Frontiers in Robotics and AI (2023)

8) Bettega J., Piva G., Richiedei D., Trevisani A., Model predictive control for path tracking in cable driven parallel robots with flexible cables: collocated vs. noncollocated control. Multibody Syst Dyn (2023)

9) Bettega J., Richiedei D., Tamellin I., Trevisani A., Integrated Inverse Dynamics and Optimized Mechanical Design in Underactuated Linear Vibratory Feeders Under Periodic Excitation. J. Vib. Eng. Technol. (2023)

10) Bettega J., Richiedei D., Tamellin I., Trevisani A., Model Inversion for Precise Path and Trajectory Tracking in an Underactuated, Non-Minimum Phase, Spatial Overhead Crane. J. Vib. Eng. Technol. (2023)

DOCTORAL COURSE IN MECHATRONICS AND PRODUCT INNOVATION ENGINEERING

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