MATLAB机器人工具箱：运动学与动力学仿真

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Tutorial

3 Manipulator kinematics

Kinematics is the study of motion without regard to the forces which cause it. Within kine-

matics one studies the position, velocity and acceleration, and all higher order derivatives of

the position variables. The kinematics of manipulators involves the study of the geometric

and time based properties of the motion, and in particular how the various links move with

respect to one another and with time.

Typical robots are serial-link manipulators comprising a set of bodies, called links, in a

chain, connected by joints

. Each joint has one degree of freedom, either translational or

rotational. For a manipulator with n joints numbered from 1 to n, there are n 1 links,

numbered from 0 to n. Link 0 is the base of the manipulator, generally ﬁxed, and link n

carries the end-effector. Joint i connects links i and i 1.

A link may be considered as a rigid body deﬁning the relationship between two neighbour-

ing joint axes. A link can be speciﬁed by two numbers, the link length and link twist, which

deﬁne the relative location of the two axes in space. The link parameters for the ﬁrst and

last links are meaningless, b

ut are arbitrarily chosen to be 0. Joints may be described by

two parameters. The link offset is the distance from one link to the next along the axis of the

joint. The joint angle is the rotation of one link with respect to the next about the joint axis.

To facilitate describing the location of each link we afﬁx a coordinate frame to it — frame i

is attached to link i. Denavit and Hartenberg[6] proposed a matrix method of systematically

assigning coordinate systems to each link of an articulated chain. The axis of revolute joint

i is aligned with z

i 1

. The x

i 1

axis is directed along the normal from z

i 1

to z

and for

intersecting axes is parallel to z

i 1

. The link and joint parameters may be summarized

as:

link length a

the offset distance between the z

i 1

and z

axes along the

axis;

link twist α

the angle from the z

i 1

axis to the z

axis about the x

axis;

link offset d

the distance from the origin of frame i 1 to the x

axis

along the z

i 1

axis;

joint angle θ

the angle between the x

i 1

and x

axes about the z

i 1

axis.

For a revolute axis θ

is the joint variable and d

is constant, while for a prismatic joint d

is variable, and θ

is constant. In many of the formulations that follow we use generalized

coordinates, q

, where

for a revolute joint

for a prismatic joint

Parallel link and serial/parallel hybrid structures are possible, though much less common in industrial manip-

ulators.

3 MANIPULATOR KINEMATICS

2. Frame i has its origin along the axis of joint i, and is frequently referred to as ‘modi-

ﬁed Denavit-Hartenberg’ (MDH) form[8]. This form is commonly used in literature

dealing with manipulator dynamics. The link transform matrix for this form differs

from (1).

Figure 8 shows the notational differences between the two forms. Note that a

is always the

length of link i, but is the displacement between the origins of frame i and frame i 1 in

one convention, and frame i

1 and frame i in the other

. The Toolbox provides kinematic

functions for both of these conventions — those for modiﬁed DH parameters are preﬁxed

by ‘m’.

3.1 Forward and inverse kinematics

For an n-axis rigid-link manipulator, the forward kinematic solution gives the coordinate

frame, or pose, of

the last link. It is obtained by repeated application of (2)

n 1

(3)

q (4)

which is the product of the coordinate frame transform matrices for each link. The pose

of the end-effector has 6 degrees of freedom in Cartesian space, 3 in translation and 3 in

rotation, so robot manipulators commonly have 6 joints or degrees of freedom to allow

arbitrary

end-effector pose. The overall manipulator transform

is frequently written as

, or T

for a 6-axis robot. The forward kinematic solution may be computed for any

manipulator, irrespective of the number of joints or kinematic structure.

Of more use in manipulator path planning is the inverse kinematic solution

T (5)

which gives the joint angles required to reach the speciﬁed end-effector position. In general

this solution is non-unique, and for some classes of manipulator no closed-form solution

xists. If the manipulator has more than 6 joints it is said to be redundant and the solution

for joint angles is under-determined. If no solution can be determined for a particular ma-

nipulator pose that conﬁguration is said to be singular. The singularity may be due to an

alignment of axes reducing the effective degrees of freedom, or the point T being out of

reach.

The manipulator Jacobian matrix, J

, transforms velocities in joint space to velocities of

the end-effector in Cartesian space. For an n

-axis manipulator the end-effector Cartesian

velocity is

˙x

˙q (6)

˙x

˙q (7)

in base or end-effector coordinates respectively

and where x is the Cartesian velocity rep-

resented by a 6-vector. For a 6-axis manipulator the Jacobian is square and provided it is

not singular can be inverted to solve for joint rates in terms of end-effector Cartesian rates.

The Jacobian will not be invertible at a kinematic singularity, and in practice will be poorly

Many papers when tabulating the ‘modiﬁed’ kinematic parameters of manipulators list a

i 1

and α

i 1

not a

and α

4 MANIPULATOR RIGID-BODY DYNAMICS

conditioned in the vicinity of the singularity, resulting in high joint rates. A control scheme

based on Cartesian rate control

˙q

˙x

(8)

was proposed by Whitney[9] and is known as resolved rate motion control. For two frames

A and B related by

n o a p the Cartesian velocity in frame A may be transformed to

frame B by

˙x

˙x (9)

where the Jacobian is given by Paul[10] as

n o a

p n p o p a

0 n o a

(10)

4 Manipulator rigid-body dynamics

Manipulator dynamics is concerned with the equations of motion, the way in which the

manipulator moves in response to torques applied by the actuators, or external forces. The

history and mathematics of the dynamics of serial-link manipulators is well covered by

Paul[1] and Hollerbach[11]. There are two problems related to manipulator dynamics that

are important to solve:

inverse dynamics in which the manipulator’s equations of motion are solved for given

motion to

determine the generalized forces, discussed further in Section 4.1, and

direct dynamics in which the equations of motion are integrated to determine the

generalized coordinate response to applied generalized forces discussed further in

Section 4.2.

The equations of motion for an n-axis manipulator are given by

Q M q ¨q C q ˙q ˙q F ˙q G q (11)

where

is the vector of generalized joint coordinates describing the pose of the manipulator

˙q is the vector of joint velocities;

¨q is the vector of joint accelerations

M is the symmetric joint-space inertia matrix, or manipulator inertia tensor

C describes Coriolis and centripetal effects — Centripetal torques are proportional to ˙q

while the Coriolis torques are proportional to ˙q

˙q

F describes viscous and Coulomb friction and is not generally considered part of the rigid-

body dynamics

G is the gravity loading

Q is the vector of generalized forces associated with the generalized coordinates q.

The equations may be derived via a number of techniques, including Lagrangian (energy

based), Newton-Euler, d’Alembert[2, 12] or Kane’s[13] method. The earliest reported work

was by Uicker[14] and Kahn[15] using the Lagrangian approach. Due to the enormous com-

putational

cost, O n

, of this approach it was not possible to compute manipulator torque

for real-time control. To achieve real-time performance many approaches were suggested,

including table lookup[16] and approximation[17, 18]. The most common approximation

was to ignore the velocity-dependent term C, since accurate positioning and high speed

motion are exclusive in typical robot applications.

4 MANIPULATOR RIGID-BODY DYNAMICS

Method Multiplications Additions For N=6

Multiply Add

Lagrangian[22] 32

25n

66,271 51,548

171

n 129

128 96

Recursive NE[22] 150n 48 131n 48 852 738

Kane[13] 646 394

Simpliﬁed RNE[25] 224 174

Table 1: Comparison of computational costs for inverse dynamics from various sources.

The last entry is achieved by symbolic simpliﬁcation using the software package ARM.

Orin et al.[19] proposed an alternative approach based on the Newton-Euler (NE) equations

of rigid-body motion applied to each link. Armstrong[20] then showed how recursion might

be applied resulting in O n complexity. Luh et al.[21] provided a recursive formulation of

the Newton-Euler equations with linear and angular velocities referred to link coordinate

frames. They suggested a time improvement from 7 9s for the Lagrangian formulation

to 4

5ms, and thus it became practical to implement ‘on-line’. Hollerbach[22] showed

how recursion could be applied to the Lagrangian form, and reduced the computation to

within a factor of 3 of the recursive NE. Silver[23] showed the equivalence of the recursive

Lagrangian and Newton-Euler forms, and that the difference in efﬁciency is due to the

representation of angular velocity.

“Kane’s equations” [13] provide another methodology for deriving the equations of motion

for a speciﬁc manipulator. A number of ‘Z’ variables are introduced, which while not

necessarily of physical signiﬁcance, lead to a dynamics

formulation with low computational

burden. Wampler[24] discusses the computational costs of Kane’s method in some detail.

The NE and Lagrange forms can be written generally in terms of the Denavit-Hartenberg

parameters — however the speciﬁc formulations, such as Kane’s, can have lower compu-

tational cost for the speciﬁc manipulator. Whilst the recursive forms are computationally

more efﬁcient, the non-recursive forms compute the individual dynamic terms (M, C and

G) directly. A comparison of computation

costs is given in Table 1.

4.1 Recursive Newton-Euler formulation

The recursive Newton-Euler (RNE) formulation[21] computes the inverse manipulator dy-

namics, that is, the joint torques required for a given set of joint angles, velocities and

accelerations. The forward recursion propagates kinematic information — such as angu-

lar velocities, angular accelerations, linear accelerations

— from the base reference frame

(inertial frame) to the end-effector. The backward recursion propagates the forces and mo-

ments exerted on each link from the end-effector of the manipulator to the base reference

frame

. Figure 9 shows the variables involved in the computation for one link.

The notation of Hollerbach[22] and Walker and Orin [26] will be used in which the left

superscript indicates the reference coordinate frame for the variable. The notation of Luh et

al.[21

] and later Lee[7, 2] is considerably less clear.

It should be noted that using MDH notation with its different axis assignment conventions the Newton Euler

formulation is expressed differently[8].

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