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# Ch. 5: Path Planning Ch. 6 Single Variable Control - PowerPoint PPT Presentation

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 PlanningCh. 6 Single Variable Control

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• Lab #3 this week

• Lab #2 writeup due 3/22

• HW #3 due today

• HW #4 distributed today

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• Recent success in the Harvard Microrobotics Lab:

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• 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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• 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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• 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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• 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

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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

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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 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 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 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 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 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 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 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 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 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 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 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 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 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 transform random excitation in case we are stuck in a local minima

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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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