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## PowerPoint Slideshow about 'Ch. 5: Path Planning Ch. 6 Single Variable Control' - derica

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Presentation Transcript

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

- 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

- 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

- 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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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 descentES159/259

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

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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

- 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

- If we also specify accelerations, we add two other parameters
- For example: quintic polynomial
- The velocity is:

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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 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

- 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

- 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

- 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

- 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

- 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

- 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

- 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

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

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Review of the Laplace transform

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