Ch 5 path planning ch 6 single variable control
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Ch. 5: Path Planning Ch. 6 Single Variable Control. Updates. Lab #3 this week Lab #2 writeup due 3/22 HW #3 due today HW #4 distributed today. Insect-sized MAVs. Recent success in the Harvard Microrobotics Lab:. Path planning overview.

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Ch 5 path planning ch 6 single variable control

Ch. 5: Path PlanningCh. 6 Single Variable Control

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Updates
Updates

  • Lab #3 this week

  • Lab #2 writeup due 3/22

  • HW #3 due today

  • HW #4 distributed today

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Insect sized mavs
Insect-sized MAVs

  • Recent success in the Harvard Microrobotics Lab:

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Path planning overview
Path planning overview

  • Want to find a path from an initial position to a final position

    • Both are defined as vectors in the configuration space

    • Initial configuration is qs and final configuration is qf

  • Instead of completely defining QO or Qfree, we develop a sequence of discrete configurations that drive the configuration from qs to qf and avoid obstacles

  • Definition:

    • g: continuous mapping from qs to qf

    • Such that:

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The relation between workspace forces and joint torques
The relation between workspace forces and joint torques

  • When we discussed velocity kinematics we said that we could use the Jacobian to describe the mapping from joint forces and torques to end effector forces and torques:

    • Jv is the velocity portion of the Jacobian (since we have only discussed workspace forces, not torques)

  • The Jacobian must be derived for each origin oi, denoted Joi

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Composing workspace forces
Composing workspace forces

  • The total joint torques acting on a manipulator is the sum of the torques from all attractive and repulsive potentials:

  • If we add workspace forces before transforming into the configuration space, we will have incorrect torques in the configuration space

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Gradient descent
Gradient descent

  • Now that we can formulate the total torques acting on the joints in the configuration space due to the artificial potentials, we can formulate a path planning algorithm

  • First, determine your initial configuration

  • Second, given a desired point in the workspace, calculate the final configuration using the inverse kinematics

    • Use this to create an attractive potential field

  • Locate obstacles in the workspace

    • Create a repulsive potential field

  • Sum the joint torques in the configuration space

  • Use gradient descent to reach your target configuration

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Gradient descent1

Algorithm:

Notation:

qi: configuration at the ith iteration

e: region of convergence

zi:influence of attractive potential for oi. Can be different for each oi

ai:defines step size of ith iteration

hi:influence of repulsive potential for oi. Can be different for each oi

r0:distance of influence for obstacles. Can be defined differently for different obstacles

Gradient descent

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Local minima

Instead we modify the gradient descent algorithm to add a random excitation in case we are stuck in a local minima

We are stuck in a local minima if successive iterations result in minimal changes in the configuration

If so, perform a random walk to get out

The random walk is defined by adding a uniformly distributed variable to each joint parameter

Local minima

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Trajectory planning
Trajectory planning random excitation in case we are stuck in a local minima

  • Either gradient descent or probabilistic roadmaps give us a sequence of way points

  • How do we smoothly connect them?

  • First, split the workspace into areas of fast and guarded motion:

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Creating smooth trajectories
Creating smooth trajectories random excitation in case we are stuck in a local minima

  • Very simple premise: fit positions and velocities to polynomials

    • The more constraints, the more terms in the polynomial

    • At least two parameters (initial and final positions)

  • We do this for each joint variable independently

  • For example: if we specify the initial and final positions and velocities, we need four parameters:

    • Cubic polynomial:

    • The velocity is:

    • This has determinant (tf- t0)4, thus we can invert and find unique solution

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Creating smooth trajectories1
Creating smooth trajectories random excitation in case we are stuck in a local minima

  • If we also specify accelerations, we add two other parameters

  • For example: quintic polynomial

    • The velocity is:

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Creating smooth trajectories2
Creating smooth trajectories random excitation in case we are stuck in a local minima

  • Therefore, for either cubic or quintic polynomials, we can use the specified parameters and solve for the a terms and create a trajectory from one way point to another

  • However, what if we want to connect multiple way points, we can do one of two things:

    • Use a quintic polynomial to describe the motion between qi and qi+1 such that:

    • This ensures that there are no jumps in velocity (and hence acceleration and motor torques)

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Creating smooth trajectories3
Creating smooth trajectories random excitation in case we are stuck in a local minima

  • However, what if we want to connect multiple way points, we can do one of two things:

    • Or we can string together multiple way points by specifying the initial and final velocities

    • E.g. three way points:

    • Thus we need an order-6 polynomial:

    • Downsides: as the number of way points increases, the order of polynomial increases

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Single variable control
Single variable control random excitation in case we are stuck in a local minima

  • How do we determine the motor/actuator inputs so as to command the end effector in a desired motion?

  • In general, the input voltage/current does not create instantaneous motion to a desired configuration

    • Due to dynamics (inertia, etc)

    • Nonlinear effects

      • Backlash

      • Friction

    • Linear effects

      • Compliance

  • Thus, we need three basic pieces of information:

    • Desired joint trajectory

    • Description of the system (ODE)

    • Measurement of actual trajectory

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Siso overview
SISO overview random excitation in case we are stuck in a local minima

  • Typical single input, single output (SISO) system:

  • We want the robot tracks the desired trajectory and rejects external disturbances

  • We already have the desired trajectory, and we assume that we can measure the actual trajectories

  • Thus the first thing we need is a system description

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Siso overview1
SISO overview random excitation in case we are stuck in a local minima

  • Need a convenient input-output description of a SISO system

  • Two typical representations for the plant:

    • Transfer function

    • State-space

  • Transfer functions represent the system dynamics in terms of the Laplace transform of the ODEs that represent the system dynamics

  • For example, if we have a 1DOF system described by:

  • We want the representation in the Laplace domain:

  • Therefore, we give the transfer function as:

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Review of the laplace transform
Review of the Laplace transform random excitation in case we are stuck in a local minima

  • Laplace transform creates algebraic equations from differential equations

  • The Laplace transform is defined as follows:

  • For example, Laplace transform of a derivative:

    • Integrating by parts:

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Review of the laplace transform1
Review of the Laplace transform random excitation in case we are stuck in a local minima

  • Similarly, Laplace transform of a second derivative:

  • Thus, if we have a generic 2nd order system described by the following ODE:

  • And we want to get a transfer function representation of the system, take the Laplace transform of both sides:

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Review of the laplace transform2
Review of the Laplace transform random excitation in case we are stuck in a local minima

  • Continuing:

  • The transient response is the solution of the above ODE if the forcing function F(t) = 0

  • Ignoring the transient response, we can rearrange:

  • This is the input-output transfer function and the denominator is called the characteristic equation

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Review of the laplace transform3
Review of the Laplace transform random excitation in case we are stuck in a local minima

  • Properties of the Laplace transform

    • Takes an ODE to a algebraic equation

    • Differentiation in the time domain is multiplication by s in the Laplace domain

    • Integration in the time domain is multiplication by 1/s in the Laplace domain

    • Considers initial conditions

      • i.e. transient and steady-state response

    • The Laplace transform is a linear operator

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Review of the laplace transform4
Review of the Laplace transform random excitation in case we are stuck in a local minima

  • for this class, we will rely on a table of Laplace transform pairs for convenience

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Review of the laplace transform5
Review of the Laplace transform random excitation in case we are stuck in a local minima

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Next class
Next class… random excitation in case we are stuck in a local minima

  • Continuation of single variable control

  • Discussion of final projects

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