Mobile Robot Predictive Trajectory Tracking

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Mobile Robot Predictive Trajectory Tracking
Martin Seyr, Stefan Jakubek
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Overview

• Problem statement
• Introduction of the mobile robot
• Outline of the general idea of the proposed
  algorithm
• Presentation of the key innovations
• Technical details
• Results
• Conclusion and outlook

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Problem statement

• Trajectory tracking accurately follow a given
  time-indexed path in the plane
• Nonlinear nonholonomic system
• Cannot be stabilized by continuous,
  time-invariant feedback
• Cannot be feedback-linearized
• Account for non-zero side-slip
• Posture stabilization and tracking using the same
  algorithm
• Most approaches deal with only one of the two
  tasks

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The mobile robot

• Cuboid with 0.075m square footprint
• 2 DC-motors driven by PWM-input voltage
• DSP executes algorithm
• mC provides sensor data and generates PWM-signal
• Dual full bridge driver amplifies control voltage

• Autonomous differential drive mini robot
  Tinyphoon, equipped with an Infineon XC167
  microcontroller and an AD BF533 digital signal
  processor

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Control concept

• Cascaded control scheme
• Inner loop velocities are controlled via state
  feedback control with integration of the control
  error
• Outer loop nonlinear predictive control using
  the closed inner loop dynamics equation for
  prediction
• Reference velocities as input into the closed
  inner loop are calculated using
  Gauss-Newton-optimization

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Key innovations
Predictive control in combination with cascade,
state space formulation of closed loop dynamics
used for prediction
Compensation of the algorithms calculation time
using an extended state space system and
predicted velocities in the control law
The inevitable side-slip during sharp turns is
accounted for by modifying the reference angle
accordingly power slide possible
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Velocity dynamics

• Linearization by
• Neglecting slip dynamics
• Effective track speed
• Least-squares-linearization of the motor
  characteristic
• State space formulation

Kinematics of the unicycle-type mobile robot
including side-slip
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Kinematics

• Unicycle-type kinematics including side-slip
• Relationship between attitude and path angle

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Control of the inner loop

• Discretization of the linear state space system
• Extended linear discrete system with integration
  of the control error
• Stabilizing effect of exact past velocities
  exploited
• State feedback control law obtained by pole
  placement or LQR-technique

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Control of the outer loop I

• Discretization of the nonlinear state space
  representation of the kinematics
• Minimization of a scalar quadratic cost function
  during every sampling interval

• To stabilize the solution and reduce the
  calculation time, shape functions are chosen

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Control of the outer loop II

• Estimation of the side-slip angle by
  Spline-interpolation

• Corrected attitude angle calculated by

• Side slip assumed to be constant in future

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Control of the outer loop III

• Minimization of the cost function by
  Taylor-series expansion

• Total derivatives of positions with respect to
  form parameters calculated recursively

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Control of the outer loop IV

• Additional terminal constraint on attitude angle
  using a linearized Lagrangian side condition
  further stabilizes the solution
• Final form of the system of equations of
  dimension 5 to be solved

• With the resulting form parameters the reference
  velocities are updated and the control law of the
  inner loop is used to calculate the PWM-inputs

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Tuning parameters

• Sampling time
• Poles of the inner loop
• Prediction horizon
• Weight matrices L and R
• Shape functions
• Number of iterations performed

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Simulation results I

• Simulation Video normal/8x slow motion
• Sharp turn at 1ms-1 track speed, side-slip angle
  over 25, minimum radius 0.27m
• Side-slip angle control power slide

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Simulation results II
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Experimental results

• Experiment Video normal speed
• Square with 0.5m side length, circle with 0.25m
  diameter, peak velocity 0.88ms-1, peak yaw rate
  12.3rads-1
• So far only wheel-encoders implemented, therefore
  no side-slip angle control possible

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Outlook and Further Work

• Implementing and testing of the adaptive extended
  Kalman filter for inertial navigation
• Adaptation of the algorithm for 4-wheeled
  autonomous vehicles
• Trajectory planning under consideration of static
  and dynamic obstacles

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Final Remarks

• Thank you for your attention!
• Further Questions?