Serial-link robot class

A concrete class that represents a serial-link arm-type robot. The mechanism is described using Denavit-Hartenberg parameters, one set per joint.

plot | display graphical representation of robot |

teach | drive the graphical robot |

isspherical | test if robot has spherical wrist |

islimit | test if robot at joint limit |

fkine | forward kinematics |

ikine6s | inverse kinematics for 6-axis spherical wrist revolute robot |

ikine3 | inverse kinematics for 3-axis revolute robot |

ikine | inverse kinematics using iterative method |

jacob0 | Jacobian matrix in world frame |

jacobn | Jacobian matrix in tool frame |

maniplty | manipulability |

jtraj | a joint space trajectory |

accel | joint acceleration |

coriolis | Coriolis joint force |

dyn | show dynamic properties of links |

fdyn | joint motion |

friction | friction force |

gravload | gravity joint force |

inertia | joint inertia matrix |

nofriction | set friction parameters to zero |

rne | joint torque/force |

payload | add a payload in end-effector frame |

perturb | randomly perturb link dynamic parameters |

links | vector of Link objects (1xN) |

gravity | direction of gravity [gx gy gz] |

base | pose of robot's base (4x4 homog xform) |

tool | robot's tool transform, T6 to tool tip (4x4 homog xform) |

qlim | joint limits, [qmin qmax] (Nx2) |

offset | kinematic joint coordinate offsets (Nx1) |

name | name of robot, used for graphical display |

manuf | annotation, manufacturer's name |

comment | annotation, general comment |

plotopt | options for plot() method (cell array) |

n | number of joints |

config | joint configuration string, eg. 'RRRRRR' |

mdh | kinematic convention boolean (0=DH, 1=MDH) |

- SerialLink is a reference object.
- SerialLink objects can be used in vectors and arrays

- Robotics, Vision & Control, Chaps 7-9, P. Corke, Springer 2011.
- Robot, Modeling & Control, M.Spong, S. Hutchinson & M. Vidyasagar, Wiley 2006.

Create a SerialLink robot object

**R** = SerialLink(**links**, **options**) is a robot object defined by a vector
of Link objects.

**R** = SerialLink(**dh**, **options**) is a robot object with kinematics defined
by the matrix **dh** which has one row per joint and each row is
[theta d a alpha] and joints are assumed revolute. An optional
fifth column sigma indicate revolute (sigma=0, default) or
prismatic (sigma=1).

**R** = SerialLink(**options**) is a null robot object with no links.

**R** = SerialLink([R1 R2 ...], **options**) concatenate robots, the base of
R2 is attached to the tip of R1.

**R** = SerialLink(**R1**, **options**) is a deep copy of the robot object **R1**,
with all the same properties.

'name', name | set robot name property |

'comment', comment | set robot comment property |

'manufacturer', manuf | set robot manufacturer property |

'base', base | set base transformation matrix property |

'tool', tool | set tool transformation matrix property |

'gravity', g | set gravity vector property |

'plotopt', po | set plotting options property |

Create a 2-link robot

L(1) = Link([ 0 | 0 a1 0], 'standard'); |

L(2) = Link([ 0 | 0 a2 0], 'standard'); |

twolink = SerialLink(L, 'name', 'two link');

Robot objects can be concatenated in two ways

R = R1 * R2; R = SerialLink([R1 R2]);

- SerialLink is a reference object, a subclass of Handle object.
- SerialLink objects can be used in vectors and arrays
- When robots are concatenated (either syntax) the intermediate base and tool transforms are removed since general constant transforms cannot be represented in Denavit-Hartenberg notation.

Manipulator forward dynamics

**qdd** = R.accel(**q**, **qd**, **torque**) is a vector (Nx1) of joint accelerations that result
from applying the actuator force/torque to the manipulator robot in state **q** and **qd**.
If **q**, **qd**, **torque** are matrices (KxN) then **qdd** is a matrix (KxN) where each row
is the acceleration corresponding to the equivalent rows of **q**, **qd**, **torque**.

**qdd** = R.accel(**x**) as above but **x**=[**q**,**qd**,**torque**].

- Uses the method 1 of Walker and Orin to compute the forward dynamics.
- This form is useful for simulation of manipulator dynamics, in conjunction with a numerical integration function.

- Efficient dynamic computer simulation of robotic mechanisms, M. W. Walker and D. E. Orin, ASME Journa of Dynamic Systems, Measurement and Control, vol. 104, no. 3, pp. 205-211, 1982.

SerialLink.rne, SerialLink, ode45

Update a robot animation

R.animate(**q**) updates an existing animation for the robot R. This will have
been created using R.plot().

Updates graphical instances of this robot in all figures.

- Not a general purpose method, used for Simulink robot animation.

Convert to string

**s** = R.char() is a string representation of the robot's kinematic parameters,
showing DH parameters, joint structure, comments, gravity vector, base and
tool transform.

Cartesian inertia matrix

**m** = R.cinertia(**q**) is the NxN Cartesian (operational space) inertia matrix which relates
Cartesian force/torque to Cartesian acceleration at the joint configuration **q**, and N
is the number of robot joints.

SerialLink.inertia, SerialLink.rne

Coriolis matrix

**C** = R.coriolis(**q**, **qd**) is the Coriolis/centripetal matrix (NxN) for
the robot in configuration **q** and velocity **qd**, where N is the number of
joints. The product **C*****qd** is the vector of joint force/torque due to velocity
coupling. The diagonal elements are due to centripetal effects and the
off-diagonal elements are due to Coriolis effects. This matrix is also
known as the velocity coupling matrix, since gives the disturbance forces
on all joints due to velocity of any joint.

If **q** and **qd** are matrices (KxN), each row is interpretted as a joint state
vector, and the result (NxNxK) is a 3d-matrix where each plane corresponds
to a row of **q** and **qd**.

**C** = R.coriolis( **qqd**) as above but the matrix **qqd** (1x2N) is [**q** **qd**].

- Joint friction is also a joint force proportional to velocity but it is eliminated in the computation of this value.
- Computationally slow, involves N^2/2 invocations of RNE.

Display parameters

R.display() displays the robot parameters in human-readable form.

- This method is invoked implicitly at the command line when the result of an expression is a SerialLink object and the command has no trailing semicolon.

SerialLink.char, SerialLink.dyn

Display inertial properties

R.dyn() displays the inertial properties of the SerialLink object in a multi-line format. The properties shown are mass, centre of mass, inertia, gear ratio, motor inertia and motor friction.

R.dyn(**J**) as above but display parameters for joint **J** only.

Integrate forward dynamics

[**T**,**q**,**qd**] = R.fdyn(**T1**, **torqfun**) integrates the dynamics of the robot over
the time interval 0 to **T** and returns vectors of time TI, joint position **q**
and joint velocity **qd**. The initial joint position and velocity are zero.
The torque applied to the joints is computed by the user function **torqfun**:

[**ti**,**q**,**qd**] = R.fdyn(**T**, **torqfun**, **q0**, **qd0**) as above but allows the initial
joint position and velocity to be specified.

The control torque is computed by a user defined function

TAU = TORQFUN(T, Q, QD, ARG1, ARG2, ...)

where **q** and **qd** are the manipulator joint coordinate and velocity state
respectively, and **T** is the current time.

[**T**,**q**,**qd**] = R.fdyn(**T1**, **torqfun**, **q0**, **qd0**, ARG1, ARG2, ...) allows optional
arguments to be passed through to the user function.

- This function performs poorly with non-linear joint friction, such as Coulomb friction. The R.nofriction() method can be used to set this friction to zero.
- If TORQFUN is not specified, or is given as 0 or [], then zero torque is applied to the manipulator joints.
- The builtin integration function ode45() is used.

SerialLink.accel, SerialLink.nofriction, SerialLink.rne, ode45

evaluate fkine for each point on a trajectory of theta_i or q_i data

Friction force

**tau** = R.friction(**qd**) is the vector of joint friction forces/torques for the
robot moving with joint velocities **qd**.

The friction model includes:

- viscous friction which is linear with velocity;
- Coulomb friction which is proportional to sign(QD).

Vector of symbolic generalized coordinates

Q = R.GENCOORDS is a vector (1xN) of symbols [q1 q2 ... qN].

[Q,QD] = R.GENCOORDS as above but QD is a vector (1xN) of symbols [qd1 qd2 ... qdN].

[Q,QD,QDD] = R.GENCOORDS as above but QDD is a vector (1xN) of symbols [qdd1 qdd2 ... qddN].

Gravity loading

**taug** = R.gravload(**q**) is the joint gravity loading for the robot in the
joint configuration **q**. Gravitational acceleration is a property of the
robot object.

If **q** is a row vector, the result is a row vector of joint torques. If
**q** is a matrix, each row is interpreted as a joint configuration vector,
and the result is a matrix each row being the corresponding joint torques.

**taug** = R.gravload(**q**, **grav**) is as above but the gravitational
acceleration vector **grav** is given explicitly.

SerialLink.rne, SerialLink.itorque, SerialLink.coriolis

default parameters for solution

Inverse kinematics for 3-axis robot with no wrist

**q** = R.ikine3(**T**) is the joint coordinates corresponding to the robot
end-effector pose **T** represented by the homogenenous transform. This
is a analytic solution for a 3-axis robot (such as the first three joints
of a robot like the Puma 560).

**q** = R.ikine3(**T**, **config**) as above but specifies the configuration of the arm in
the form of a string containing one or more of the configuration codes:

'l' | arm to the left (default) |

'r' | arm to the right |

'u' | elbow up (default) |

'd' | elbow down |

- The same as IKINE6S without the wrist.
- The inverse kinematic solution is generally not unique, and depends on the configuration string.
- Joint offsets, if defined, are added to the inverse kinematics to generate Q.

Inverse kinematics for a PUMA 560 based on the equations by Paul and Zhang From The International Journal of Robotics Research Vol. 5, No. 2, Summer 1986, p. 32-44

Robert Biro with Gary Von McMurray, GTRI/ATRP/IIMB, Georgia Institute of Technology 2/13/95

SerialLink.FKINE, SerialLink.IKINE

Inverse kinematics for 6-axis robot with spherical wrist

**q** = R.ikine6s(**T**) is the joint coordinates corresponding to the robot
end-effector pose **T** represented by the homogenenous transform. This
is a analytic solution for a 6-axis robot with a spherical wrist (such as
the Puma 560).

**q** = R.IKINE6S(**T**, **config**) as above but specifies the configuration of the arm in
the form of a string containing one or more of the configuration codes:

'l' | arm to the left (default) |

'r' | arm to the right |

'u' | elbow up (default) |

'd' | elbow down |

'n' | wrist not flipped (default) |

'f' | wrist flipped (rotated by 180 deg) |

- Only applicable for an all revolute 6-axis robot RRRRRR.
- The inverse kinematic solution is generally not unique, and depends on the configuration string.
- Joint offsets, if defined, are added to the inverse kinematics to generate Q.

- Inverse kinematics for a PUMA 560, Paul and Zhang, The International Journal of Robotics Research, Vol. 5, No. 2, Summer 1986, p. 32-44

Robert Biro with Gary Von McMurray, GTRI/ATRP/IIMB, Georgia Institute of Technology 2/13/95

SerialLink.FKINE, SerialLink.IKINE

Manipulator inertia matrix

**i** = R.inertia(**q**) is the symmetric joint inertia matrix (NxN) which relates
joint torque to joint acceleration for the robot at joint configuration **q**.

If **q** is a matrix (KxN), each row is interpretted as a joint state
vector, and the result is a 3d-matrix (NxNxK) where each plane corresponds
to the inertia for the corresponding row of **q**.

- The diagonal elements I(J,J) are the inertia seen by joint actuator J.
- The off-diagonal elements I(J,K) are coupling inertias that relate acceleration on joint J to force/torque on joint K.
- The diagonal terms include the motor inertia reflected through the gear ratio.

SerialLink.RNE, SerialLink.CINERTIA, SerialLink.ITORQUE

Joint limit test

**v** = R.islimit(**q**) is a vector of boolean values, one per joint,
false (0) if **q**(i) is within the joint limits, else true (1).

- Joint limits are purely advisory and are not used in any other function. Just seemed like a useful thing to include...

Test for spherical wrist

R.isspherical() is true if the robot has a spherical wrist, that is, the last 3 axes are revolute and their axes intersect at a point.

Inertia torque

**taui** = R.itorque(**q**, **qdd**) is the inertia force/torque vector (1xN) at the
specified joint configuration **q** (1xN) and acceleration **qdd** (1xN), that is,
**taui** = INERTIA(**q**)***qdd**.

If **q** and **qdd** are matrices (KxN), each row is interpretted as a joint state
vector, and the result is a matrix (KxN) where each row is the corresponding
joint torques.

- If the robot model contains non-zero motor inertia then this will included in the result.

SerialLink.rne, SerialLink.inertia

Jacobian in world coordinates

**j0** = R.jacob0(**q**, **options**) is the Jacobian matrix (6xN) for the robot in
pose **q** (1xN). The manipulator Jacobian matrix maps joint velocity to
end-effector spatial velocity V = **j0***QD expressed in the world-coordinate
frame.

'rpy' | Compute analytical Jacobian with rotation rate in terms of roll-pitch-yaw angles |

'eul' | Compute analytical Jacobian with rotation rates in terms of Euler angles |

'trans' | Return translational submatrix of Jacobian |

'rot' | Return rotational submatrix of Jacobian |

- The Jacobian is computed in the world frame and transformed to the end-effector frame.
- The default Jacobian returned is often referred to as the geometric Jacobian, as opposed to the analytical Jacobian.

SerialLink.jacobn, jsingu, deltatr, tr2delta, jsingu

Derivative of Jacobian

**jdq** = R.jacob_dot(**q**, **qd**) is the product (6x1) of the derivative of the
Jacobian (in the world frame) and the joint rates.

- Useful for operational space control XDD = J(Q)QDD + JDOT(Q)QD
- Written as per the text and not very efficient.

- Fundamentals of Robotics Mechanical Systems (2nd ed) J. Angleles, Springer 2003.

: SerialLink.jacob0, diff2tr, tr2diff

Jacobian in end-effector frame

**jn** = R.jacobn(**q**, **options**) is the Jacobian matrix (6xN) for the robot in
pose **q**. The manipulator Jacobian matrix maps joint velocity to
end-effector spatial velocity V = **jn***QD in the end-effector frame.

'trans' | Return translational submatrix of Jacobian |

'rot' | Return rotational submatrix of Jacobian |

- This Jacobian is often referred to as the geometric Jacobian.

Differential Kinematic Control Equations for Simple Manipulators, Paul, Shimano, Mayer, IEEE SMC 11(6) 1981, pp. 456-460

SerialLink.jacob0, jsingu, delta2tr, tr2delta

Joint space trajectory

**q** = R.jtraj(**T1**, **t2**, **k**) is a joint space trajectory (KxN) where the joint
coordinates reflect motion from end-effector pose **T1** to **t2** in **k** steps with
default zero boundary conditions for velocity and acceleration.
The trajectory **q** has one row per time step, and one column per joint,
where N is the number of robot joints.

- Requires solution of inverse kinematics. R.ikine6s() is used if appropriate, else R.ikine(). Additional trailing arguments to R.jtraj() are passed as trailing arugments to these functions.

jtraj, SerialLink.ikine, SerialLink.ikine6s

Manipulability measure

**m** = R.maniplty(**q**, **options**) is the manipulability index measure for the robot
at the joint configuration **q**. It indicates dexterity, that is, how isotropic
the robot's motion is with respect to the 6 degrees of Cartesian motion.
The measure is high when the manipulator is capable of equal motion in all
directions and low when the manipulator is close to a singularity.

If **q** is a matrix (MxN) then **m** (Mx1) is a vector of manipulability
indices for each pose specified by a row of **q**.

[**m**,**ci**] = R.maniplty(**q**, **options**) as above, but for the case of the Asada
measure returns the Cartesian inertia matrix **ci**.

Two measures can be selected:

- Yoshikawa's manipulability measure is based on the shape of the velocity ellipsoid and depends only on kinematic parameters.
- Asada's manipulability measure is based on the shape of the acceleration ellipsoid which in turn is a function of the Cartesian inertia matrix and the dynamic parameters. The scalar measure computed here is the ratio of the smallest/largest ellipsoid axis. Ideally the ellipsoid would be spherical, giving a ratio of 1, but in practice will be less than 1.

'T' | manipulability for transational motion only (default) |

'R' | manipulability for rotational motion only |

'all' | manipulability for all motions |

'dof', D | D is a vector (1x6) with non-zero elements if the corresponding DOF is to be included for manipulability |

'yoshikawa' | use Yoshikawa algorithm (default) |

'asada' | use Asada algorithm |

- By default the measure includes rotational and translational dexterity, but this involves adding different units. It can be more useful to look at the translational and rotational manipulability separately.

- Analysis and control of robot manipulators with redundancy, T. Yoshikawa, Robotics Research: The First International Symposium (M. Brady and R. Paul, eds.), pp. 735-747, The MIT press, 1984.
- A geometrical representation of manipulator dynamics and its application to arm design, H. Asada, Journal of Dynamic Systems, Measurement, and Control, vol. 105, p. 131, 1983.

SerialLink.inertia, SerialLink.jacob0

Concatenate robots

R = R1 * R2 is a robot object that is equivalent to mechanically attaching robot R2 to the end of robot R1.

- If R1 has a tool transform or R2 has a base transform these are discarded since DH convention does not allow for arbitrary intermediate transformations.

Remove friction

**rnf** = R.nofriction() is a robot object with the same parameters as R but
with non-linear (Coulomb) friction coefficients set to zero.

**rnf** = R.nofriction('all') as above but all friction coefficients set to zero.

**rnf** = R.nofriction('viscous') as above but only viscous friction coefficients
are set to zero.

- Non-linear (Coulomb) friction can cause numerical problems when integrating the equations of motion (R.fdyn).
- The resulting robot object has its name string prefixed with 'NF/'.

SerialLink.fdyn, Link.nofriction

Add payload mass

R.payload(**m**, **p**) adds a payload with point mass **m** at position **p**
in the end-effector coordinate frame.

SerialLink.rne, SerialLink.gravload

Perturb robot parameters

**rp** = R.perturb(**p**) is a new robot object in which the dynamic parameters (link
mass and inertia) have been perturbed. The perturbation is multiplicative so
that values are multiplied by random numbers in the interval (1-**p**) to (1+**p**).
The name string of the perturbed robot is prefixed by '**p**/'.

Useful for investigating the robustness of various model-based control schemes. For example to vary parameters in the range +/- 10 percent is:

r2 = p560.perturb(0.1);

Graphical display and animation

R.plot(**q**, **options**) displays a graphical animation of a robot based on
the kinematic model. A stick figure polyline joins the origins of
the link coordinate frames. The robot is displayed at the joint angle **q** (1xN), or
if a matrix (MxN) it is animated as the robot moves along the M-point trajectory.

'workspace', W | size of robot 3D workspace, W = [xmn, xmx ymn ymx zmn zmx] |

'delay', d | delay betwen frames for animation (s) |

'fps', fps | set number of frames per second for display |

'[no]loop' | loop over the trajectory forever |

'mag', scale | annotation scale factor |

'cylinder', C | color for joint cylinders, C=[r g b] |

'ortho' | orthogonal camera view (default) |

'perspective' | perspective camera view |

'xyz' | wrist axis label is XYZ |

'noa' | wrist axis label is NOA |

'[no]raise' | autoraise the figure (very slow). |

'[no]render' | controls shaded rendering after drawing |

'[no]base' | controls display of base 'pedestal' |

'[no]wrist' | controls display of wrist |

'[no]shadow' | controls display of shadow |

'[no]name' | display the robot's name |

'[no]jaxes' | control display of joint axes |

'[no]joints' | controls display of joints |

'movie', M | save frames as files in the folder M |

The **options** come from 3 sources and are processed in order:

- Cell array of options returned by the function PLOTBOTOPT (if it exists)
- Cell array of options given by the 'plotopt' option when creating the SerialLink object.
- List of arguments in the command line.

Many boolean **options** can be enabled or disabled with the 'no' prefix. The
various option sources can toggle an option, the last value is taken.

To save the effort of processing **options** on each call they can be preprocessed by

opts = robot.plot({'opt1', 'opt2', ... });

and the resulting object can be passed in to subsequent calls instead of text-based
**options**, for example:

robot.plot(q, opts);

The robot is displayed as a basic stick figure robot with annotations such as:

- shadow on the floor
- XYZ wrist axes and labels
- joint cylinders and axes

which are controlled by **options**.

The size of the annotations is determined using a simple heuristic from the workspace dimensions. This dimension can be changed by setting the multiplicative scale factor using the 'mag' option.

- If no figure exists one will be created and teh robot drawn in it.
- If no robot of this name is currently displayed then a robot will be drawn in the current figure. If hold is enabled (hold on) then the robot will be added to the current figure.
- If the robot already exists then that graphical model will be found and moved.

If one or more plots of this robot already exist then these will all
be moved according to the argument **q**. All robots in all windows with
the same name will be moved.

Create a robot in figure 1

figure(1) p560.plot(qz);

Create a robot in figure 2

figure(2) p560.plot(qz);

Now move both robots

p560.plot(qn)

Multiple robots can be displayed in the same plot, by using "hold on" before calls to robot.plot().

Create a robot in figure 1

figure(1) p560.plot(qz);

Make a clone of the robot named bob

bob = SerialLink(p560, 'name', 'bob');

Draw bob in this figure

hold on bob.plot(qn)

To animate both robots so they move together:

qtg = jtraj(qr, qz, 100); for q=qtg'

p560.plot(q'); bob.plot(q');

end

- The 'movie' options saves frames as files NNNN.png.
- When using 'movie' option ensure that the window is fully visible.
- To convert frames to a movie use a command like:

ffmpeg -r 10 -i %04d.png out.avi

- Delay betwen frames can be eliminated by setting option 'delay', 0 or 'fps', Inf.
- By default a quite detailed plot is generated, but turning off labels, axes, shadows etc. will speed things up.
- Each graphical robot object is tagged by the robot's name and has UserData that holds graphical handles and the handle of the robot object.
- The graphical state holds the last joint configuration which can be retrieved using q = robot.plot().

plotbotopt, SerialLink.animate, SerialLink.fkine

a cell array of options and return a struct

Inverse dynamics

**tau** = R.rne(**q**, **qd**, **qdd**) is the joint torque required for the robot R
to achieve the specified joint position **q**, velocity **qd** and acceleration **qdd**.

**tau** = R.rne(**q**, **qd**, **qdd**, **grav**) as above but overriding the gravitational
acceleration vector in the robot object R.

**tau** = R.rne(**q**, **qd**, **qdd**, **grav**, **fext**) as above but specifying a wrench
acting on the end of the manipulator which is a 6-vector [Fx Fy Fz Mx My Mz].

**tau** = R.rne(**x**) as above where **x**=[**q**,**qd**,**qdd**].

**tau** = R.rne(**x**, **grav**) as above but overriding the gravitational
acceleration vector in the robot object R.

**tau** = R.rne(**x**, **grav**, **fext**) as above but specifying a wrench
acting on the end of the manipulator which is a 6-vector [Fx Fy Fz Mx My Mz].

[**tau**,**wbase**] = R.rne(**x**, **grav**, **fext**) as above but the extra output is the
wrench on the base.

If **q**,**qd** and **qdd** (MxN), or **x** (Mx3N) are matrices with M rows representing a
trajectory then **tau** (MxN) is a matrix with rows corresponding to each trajectory
step.

- The robot base transform is ignored.
- The torque computed contains a contribution due to armature inertia and joint friction.
- RNE can be either an M-file or a MEX-file.
- See the README file in the mex folder for details on how to configure MEX-file operation.
- The M-file is a wrapper which calls either RNE_DH or RNE_MDH depending on the kinematic conventions used by the robot object.
- Currently the MEX-file version does not compute WBASE.

SerialLink.accel, SerialLink.gravload, SerialLink.inertia

Graphical teach pendant

R.teach(**options**) drive a graphical robot by means of a graphical slider panel.
If no graphical robot exists one is created in a new window. Otherwise
all current instances of the graphical robot are driven.

**h** = R.teach(**options**) as above but returns a handle for the teach window. Can
be used to programmatically delete the teach window.

'eul' | Display tool orientation in Euler angles |

'rpy' | Display tool orientation in roll/pitch/yaw angles |

'approach' | Display tool orientation as approach vector (z-axis) |

'degrees' | Display angles in degrees (default radians) |

'q0', q | Set initial joint coordinates |

- The record button adds the current joint coordinates as a row to the robot's qteach property.
- The Quit button destroys the teach window.

- The slider limits are derived from the joint limit properties. If not set then for
- a revolute joint they are assumed to be [-pi, +pi]
- a prismatic joint they are assumed unknown and an error occurs.

on changes to a slider or to the edit box showing joint coordinate

src | the object that caused the event |

name | name of the robot |

j | the joint index concerned (1..N) |

slider | true if the |

Robotics Toolbox for MATLAB © 1990-2012 Peter Corke.