# Manipulability ellipsoid matlab

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4-RRR Planar Parallel Manipulator Manipulability Ellipsoid at Workspace Boundary

The MATLAB codes show simple examples for a manipulability tracking task, where the robot is required to track a desired manipulability ellipsoid either as its main task or as a secondary objective where the nullspace of the robot is exploited.

This approach offers the possibility of transferring posture-dependent task requirements such as preferred directions for motion and force exertion in operational space, which are encapsulated in manipulability ellipsoids. The proposed formulation exploits tensor-based representations and takes into account that manipulability ellipsoids lie on the manifold of symmetric positive definite matrices.

The proposed mathematical development is compatible with statistical methods providing 4th-order covariances see [1]which are here exploited to reflect the tracking precision required by the task. Skip to content. Dismiss Join GitHub today GitHub is home to over 40 million developers working together to host and review code, manage projects, and build software together.

## MANIPULABILITY INDEX OF A PARALLEL ROBOT MANIPULATOR

Find file. Sign in Sign up. Go back. Launching Xcode If nothing happens, download Xcode and try again. Latest commit Fetching latest commitâ€¦. ManipulabilityTracking The MATLAB codes show simple examples for a manipulability tracking task, where the robot is required to track a desired manipulability ellipsoid either as its main task or as a secondary objective where the nullspace of the robot is exploited.

The user can: 1. Use different controller gains for the manipulability tracking 2.

Change the initial conditions and desired manipulability ellipsoid 3. The matrix gain used for the manipulability tracking controller is now defined as the inverse of a 2nd-order covariance matrix representing the variability information obtained from a learning algorithm see [1]. Change the number of iterations 2. Choose different values for the covariance-based controller gain 3.

Modify the robot kinematics by using the Robotics Toolbox functionalities 4. Here, the robot is required to hold a desired Cartesian position as main task, while matching a desired manipulability ellipsoid as secondary task using the manipulability Jacobian formulation Mandel notation. Use different controller gains for both the position controller and the manipulability tracking 2. Change the initial conditions, desired Cartesian position and desired manipulability ellipsoid 3.

Calinon, S. IEEE Intl. You signed in with another tab or window.But even if a configuration is nonsingular, it still may be close to being singular. The manipulability ellipsoid we saw in the first video of this chapter is one way to visualize how close a robot is to being singular.

As the second joint angle approaches zero, the ellipse squashes in the direction it is difficult to move and stretches in the orthogonal direction, until, at the singularity, the ellipse collapses to a line segment.

As we will see shortly, we can assign a measure of just how close the robot is to being singular according to how close the ellipse is to collapsing. The robot has n joints, so the Jacobian is an m by n matrix. A sphere of joint velocities, like the circle shown here, is defined by the equation theta-dot transpose times theta-dot equals 1.

If we assume the Jacobian is invertible, which is not strictly necessary, then we can rewrite the equation as shown here. Rearranging, we get this, and rearranging once more, we get this.

The A matrix is both symmetric and positive definite, and so is its inverse. It is well known that the quadratic equation x-transpose times A-inverse times x equals 1 defines an ellipsoid of x values that satisfy the equation.

In general, this ellipsoid is an m-minusdimensional surface in the m-dimensional space of x, but this figure shows the case where x is a 3-vector. The principal axes of the ellipsoid are aligned with the eigenvectors of A and the half-lengths of the ellipsoid along the principal axes are the square roots of the eigenvalues.

This figure shows the manipulability ellipsoid and the force ellipsoid for a 2R robot at a particular configuration. Since the matrix defining the manipulability ellipsoid is just the inverse of the matrix defining the force ellipsoid, the two ellipsoids have the same principal axes, and the lengths of the principal semi-axes are just the reciprocals of each other.

In other words, only small forces can be applied in directions where large velocities can be attained, and only small velocities are possible in directions where large forces can be applied. Now that we can visualize the end-effector motion capabilities as a manipulability ellipsoid, we can assign a single number representing how close the robot is to being singular.

These numbers are called manipulability measures.

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The first manipulability measure is the ratio of the longest axis to the shortest axis of the ellipsoid.

This measure is lower-bounded by 1, and if it is equal to 1, we say that the manipulability ellipsoid is isotropic; it is equally easy to move in any direction. On the other hand, as the robot approaches a singularity, this number grows large.

The second measure is just the square of the first measure, often called the condition number of the matrix A. A final measure is the square root of the product of the eigenvalues of A, which is proportional to the volume of the manipulability ellipsoid. If the manipulability ellipsoid volume becomes large, then the force ellipsoid volume becomes small, and vice-versa. Finally, consider the case that the Jacobian corresponds to the body Jacobian derived in this chapter.

The 6-by-n body Jacobian can be split into the 3-by-n angular velocity Jacobian J-b-omega and the 3-by-n linear velocity Jacobian J-b-v. This separation into linear and angular components is useful, because the units of the angular velocity and linear velocity are different. So, this concludes Chapter 5. You should now have a solid understanding of how to derive and interpret the Jacobian, a fundamental object in robotics that is heavily used in many robot motion planners and controllers.By using our site, you acknowledge that you have read and understand our Cookie PolicyPrivacy Policyand our Terms of Service.

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Ellipse article on Wikipedia had a simple JavaScript code to draw ellipses. The answers from Jacob and Amro are very good examples for computing and plotting points for an ellipse. I'll address some easy ways you can plot an ellipsoid The following creates the matrices xyand z for an ellipsoid centered at the origin with semi-axis lengths of 4, 2, and 1 for the x, y, and z directions, respectively:.

You can then use the function MESH to plot it, returning a handle to the plotted surface object:. The following applies a rotation of 45 degrees around the y-axis:. Also, if you want to use the rotated plot points for further calculations, you can get them from the plotted surface object:. Create two vectors, one of the x-coordinates of the points of the circumference of the ellipsoid, one of the y-coordinates.

Make these vectors long enough to satisfy your accuracy requirements. Plot the two vectors as x,y pairs joined up. I'd drop the for loops from your code, much clearer if you use vector notation. Also I'd format your question using the SO markup for code to make it all clearer to your audience. Ellipse article on Wikipedia and Rotation matrix. However, this is simple solution is good to have a quick glance how the ellipse looks like.

If you want to have a nice plot, look at the other solutions. I don't know in which version of Matlab this was added, but it's available at least from version Rb on. Learn more.

Asked 10 years, 2 months ago. Active 10 months ago.

The simplest way might be to use the Matlab function pdeellip xc,yc,a,b,phi For example: pdeellip 0,0,1,0. Correct, but that was not asked by ooi.

Stack Overflow for Teams is a private, secure spot for you and your coworkers to find and share information. Ellipse article on Wikipedia had a simple JavaScript code to draw ellipses.

The answers from Jacob and Amro are very good examples for computing and plotting points for an ellipse. I'll address some easy ways you can plot an ellipsoid The following creates the matrices xyand z for an ellipsoid centered at the origin with semi-axis lengths of 4, 2, and 1 for the x, y, and z directions, respectively:.

You can then use the function MESH to plot it, returning a handle to the plotted surface object:. The following applies a rotation of 45 degrees around the y-axis:. Also, if you want to use the rotated plot points for further calculations, you can get them from the plotted surface object:. Create two vectors, one of the x-coordinates of the points of the circumference of the ellipsoid, one of the y-coordinates.

Make these vectors long enough to satisfy your accuracy requirements. Plot the two vectors as x,y pairs joined up. I'd drop the for loops from your code, much clearer if you use vector notation. Also I'd format your question using the SO markup for code to make it all clearer to your audience. Ellipse article on Wikipedia and Rotation matrix. However, this is simple solution is good to have a quick glance how the ellipse looks like.

If you want to have a nice plot, look at the other solutions. I don't know in which version of Matlab this was added, but it's available at least from version Rb on. Learn more. Asked 10 years, 2 months ago. Active 10 months ago. Viewed 94k times. Matthew Simoneau 5, 6 6 gold badges 32 32 silver badges 45 45 bronze badges.

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Active Oldest Votes. Amro Amro k 23 23 gold badges silver badges bronze badges. Glorfindel Jacob Jacob Sardar Usama Documentation Help Center. A referenceEllipsoid object encapsulates a reference ellipsoid, modeled as an oblate spheroid with three additional properties: name, unit of length of the semi-major and semi-minor axes, and a numerical EPSG code.

You can create a general referenceEllipsoid object with the referenceEllipsoid function described here. You can also create a referenceEllipsoid with properties specific to the World Geodetic System reference ellipsoid using the wgs84Ellipsoid function. The values of the SemimajorAxis and SemiminorAxis properties are in meters. All of the nearly 60 codes in the EPSG ellipsoid table are supported.

The unit of length used for the SemimajorAxis and SemiminorAxis properties depends on the ellipsoid selected, and is indicated in the property LengthUnit.

The unit of length can be any length unit supported by the validateLengthUnit function. Numerical EPSG code, specified as an empty vector or an integer between andalthough not all integers in this range are valid numerical EPSG codes. When the reference ellipsoid represents the unit sphere, Code is an empty vector, [].

Name of the reference ellipsoid, specified as a character vector. Both the short version and the long version of the ellipsoid name are acceptable as values of the Name property. When the reference ellipsoid represents the unit sphere, Name is the character vector 'Unit Sphere'. Example: 'World Geodetic System '. Unit of length for the ellipsoid axes, specified as a character vector. The character vector can be empty, or it can be any unit of length accepted by the validateLengthUnit function.

When the reference ellipsoid represents the unit sphere, LengthUnit is the empty character vector ''. Equatorial radius of ellipsoid, specified as a positive, finite scalar. The SemimajorAxis property is expressed in units of length specified by LengthUnit.

When the SemimajorAxis property is changed, the SemiminorAxis property scales as needed to preserve the shape of the ellipsoid and the values of shape-related properties including InverseFlattening and Eccentricity.

The only way to change the SemimajorAxis property is to set it directly, using dot notation. Distance from center of ellipsoid to pole, specified as a nonnegative, finite scalar. The value of SemiminorAxis is always less than or equal to SemimajorAxisand is expressed in units of length specified by LengthUnit.

When the SemiminorAxis property is changed, the SemimajorAxis property remains unchanged, but the shape of the ellipsoid changes, which is reflected in changes in the values of InverseFlatteningEccentricityand other shape-related properties. Reciprocal of flattening, specified as positive scalar in the range [1, Inf ]. When the InverseFlattening property is changed, other shape-related properties update, including Eccentricity.Sign in to comment.

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Vote 0. Commented: Image Analyst on 6 Jan Create an AI-powered research feed to stay up to date with new papers like this posted to ArXiv.