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## Research Topics

# Trajectory tracking and optimal control

for mechanical systems with unilateral constraints

When a robotic system interacts at relatively high speed with a rigid obstacle, impulsive forces are exchanged. To model, analyze, and control the dynamic behavior of these systems, one can make use of the theory of nonsmooth mechanics: namely, we model these robotics systems as mechanical systems with unilateral position constraints. This research project addresses the problem of trajectory tracking and motion planning of systems with unilateral constraints using both simulation studies and real experiments. Connections with the problem of achieving stable bipedal locomotion will also be drawn.

Using standard PD feedback control to control a system with unilateral constraints can result in undesired behaviors. The poor performance of the standard PD controller is the result of the inedequacy of the standard notion of tracking error. By defining a new notion of tracking error, this problem can be overcome as shown in the simulation results given below. This new control strategy emerged from the sensitivity analysis of hybrid system with state jumps, It will also be used as the building block for developing numerical optimal control algorithms for exploring the trajectory manifold of mechanical systems with unilateral constraints.

*
Trajectory tracking of a controlled bouncing mass. The plot on the right shows the poor performance (red)
of the standard PD controller plus acceleration feedforward
for tracking of nominal trajectory with impacts (blue). The plot on the left shows a significant
improvement (black) due the hybrid PD controller plus acceleration feedforward that makes use
of the extended ante- and post-event trajectories. In both cases, the proportional and derivative
gains are identical.
*

For further reading:

**
Sensitivity analysis of hybrid systems with state jumps with application to trajectory tracking**

A. Saccon, N. van de Wouw, H. Nijmeijer

* Decision and Control (CDC), 2014 IEEE 53rd Annual Conference on, pages 3065-3070 *

**On Optimal Trajectory Tracking for Mechanical Systems with Unilateral Constraints, **

M Rijnen, A Saccon, H Nijmeijer,

*IEEE Conference on Decision and Control (CDC), Japan, 2015*

**Trajectory Tracking of Mechanical Systems with Unilateral Constraints:
Experimental Results of a Recently Introduced Hybrid PD Feedback Controller, **

GP Incremona, A Saccon, A Ferrara, H Nijmeijer,

*IEEE Conference on Decision and Control (CDC), Japan, 2015*

# Energy-Optimal Multiple Vehicle Motion Planning

Motion planning of autonomous vehicles in the presence of geometric, temporal, and energy-related constraints is a challenging problem that is at the core of many real applications. In fact, the rapid development of the field of robotics in the past decade, going from single robot tasks to missions that require coordination, cooperation, and communication among a number of networked vehicles makes the availability of versatile motion planners increasingly important.

We tackle the problem of multiple vehicle motion planning by adopting an optimal control theoretical setting and including explicitly in the problem formulation the possibly complex dynamics of the vehicles involved. By doing so, the planning tool that we develop goes one step further in that it affords mission designers an expert manner to understand how different cost functions and vehicle characteristics impact on the expected performance of a group of vehicles.

*
An example of solution for motion planning in a 3D environment,
an example with collision and obstacle avoidance in the plane,
and a terrain based navigation example
*

For further reading:

**
Energy-Optimal Motion Planning for Multiple Robotic Vehicles With Collision Avoidance, **

AJ Haeusler, A Saccon, AP Aguiar, J Hauser, AM Pascoal,

*IEEE Transactions on Control Systems Technology (accepted, available online), 2016*

**
Constrained Motion Planning for Multiple Vehicles on SE(3), **

A. Saccon, A.P. Aguiar, A.J. Husler, J. Hauser, and A. Pascoal,

51st IEEE Conference on Decision and Control, Maui, Hawaii, USA, Dec. 2012

**
Cooperative Motion Planning for Multiple Autonomous Marine Vehicles, **

A.J. Husler, A. Saccon, A.P. Aguiar, J. Hauser, and A. Pascoal

9th IFAC Conference on Manoeuvring and Control of Marine Craft (MCMC), Arenzano, Italy, Sep. 2012.

# Numerical algorithms for solving standard and

geometric optimal control problems

The optimal control of a continuous time process is among the oldest and most extensively studied problems in control theory. The main pillars of optimal control theory are Bellman`s principle of optimality and Pontryagin`s maximum principle, both developed during the 60`s, and the Hamilton-Jacobi-Bellman partial differential equation and its unique viscosity solution, studied deeply in the 80`s.

The research focuses on the development of numerical algorithms
for solving nonlinear optimal control problems for dynamical systems
evolving in R^{n} as well as on particular smooth manifolds (Lie groups).
The algorithm that we developed is a direct method for
solving continuous time optimal control problems, generating
a descending sequence of system trajectories. In contrast to
many direct methods, however, the continuous-time optimal control
problem is not transcribed into a discrete optimization problem,
but rather a continuous-time second-order approximation
is computed at each iteration.

*
A graphical representation of the algorithmic steps at each iteration and an example of the convergence rate plot.
*

For further reading:

A. Saccon, J. Hauser, A.P. Aguiar

**
Optimal Control on Lie Groups: The Projection Operator Approach**

* IEEE Transactions on Automatic Control ,
Volume 58, Issue 9, Sept. 2013
*

*
Lie Group Projection Operator Approach: Optimal Control on TSO(3),
A. Saccon, A.P. Aguiar, J. Hauser,
50th IEEE Conference on Decision and Control (CDC), Orlando, FL, pages 6973-6978, 2011
*

**
Optimal Control on Lie Groups: Implementations Details of the Projection Operator Approach, **

A. Saccon, J. Hauser, A.P. Aguiar,

18th IFAC World Congress, Milan, Italy, pages 14567-14572, 2011

**
Optimal Control on Non-Compact Lie Groups: A Projection Operator Approach, **

A. Saccon, J. Hauser, A.P. Aguiar,

49th IEEE Conference on Decision and Control, Atlanta, Georgia, USA, pages 7111-7116, 2010

**
Exploration of Kinematic Optimal Control on the Lie Group SO(3), **

A. Saccon, J. Hauser, A.P. Aguiar,

8th IFAC Symposium on Nonlinear Control Systems, Bologna, Italy, pages 1302-1307, 2010

**
A Barrier Function Method for the Optimization of Trajectory Functionals with Constraints, **

J. Hauser, A. Saccon,

45rd IEEE Conference on Decision and Control(CDC), San Diego, CA, USA. December 13-15, 2006.

# A virtual rider for racing motorcycles

Automotive industry relies increasingly on simulation tools for the analysis of the performance of new vehicles. These tools integrate multibody dynamics together with finite element methods and optimization packages to compute performance limits, forces, strains, and fatigue. The simulated vehicles are called virtual prototypes, and the tools, virtual prototyping tools. A fundamental characteristic of virtual prototyping is the possibility of simulating standard maneuvers for the dynamic analysis of vehicles as dictated by international norms. This implies the possibility of driving the virtual vehicle exactly as if it were real, by acting on throttle, brakes, clutch, gears and steering, just as a real driver would do.

We proposed an elegant and effective solution to ride a multibody motorcycle model along a desired ground path with specified velocity profile. The controller relies on maneuver regulation techniques and has been implemented on a commercial simulation software. We also developed a numerical technique to compute the mintime velocity profile for a given vehicle and race track.

*
A graphical representation of the longitudinal and transverse coordiantes used
in the maneuver regulation controller and the user interface in ADAMS and VI-Motorcycle.
*

For further reading:

**
A Virtual Rider for Motorcycles: Maneuver Regulation of a Multi-Body Vehicle Model **

A. Saccon, A. Beghi, J. Hauser

* IEEE Transactions on Control Systems Technology*,
Volume 21, Issue 2, March 2013

**
Trajectory Exploration of a Rigid Motorcycle Model**

A. Saccon, J. Hauser, A. Beghi

* IEEE Transactions on Control Systems Technology*,
Volume 20, Issue 2, March 2012

**
Motorcycle Modeling for High-Performance Maneuvering. Maximum velocity profile estimation, **

J. Hauser, A. Saccon,

IEEE Control Systems Magazine, volume 26, issue 5, pages 89-105, 2006

**
On the Driven Inverted Pendulum, **

J. Hauser,A. Saccon, R.Frezza,

44rd IEEE Conference on Decision and Control (CDC), Seville, Spain, December 12-15, 2005.

**
Quasi-Steady-State Approximation of the Dynamics of a Planar Motorcycle, **

A. Saccon, J. Hauser,

43rd IEEE Conference on Decision and Control (CDC), Atlantis, Paradise Island, Bahamas, 2004.

Last updated: December 21, 2015