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

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next class
Next class…
  • Continuation of single variable control
  • Discussion of final projects

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