MATLAB机器人工具箱：计算与仿真

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"Robotics TOOLBOX(英文版)" 是一个专用于MATLAB的机器人软件工具包，由澳大利亚开发，主要用于进行机器人的运动学和动力学计算。它支持诸如Kinematics、Dynamics以及轨迹生成等机器人领域的多种功能，适用于模拟及真实机器人实验结果的分析。正文： Robotics TOOLBOX for MATLAB是一款在MATLAB环境中运行的软件工具，其最新版本为Release 7.1。这个工具包由Peter I. Corke开发，并提供了对Puma560等机器人的示例模型。Corke博士是来自澳大利亚CSIRO Manufacturing Science and Technology的专家，他在机器人领域有深厚的造诣，并维护着相关的网站和邮件列表，为用户提供支持和资源。工具包的主要功能包括： 1. **运动学计算**：该工具箱支持Denavit-Hartenberg (DH) 参数表示法，这是一种广泛用于描述机器人关节和连杆之间相对位置和姿态的方法。用户可以轻松计算机器人的前向和逆向运动学，即从关节变量（如角度q）到笛卡尔坐标系中的位置和姿态，反之亦然。 2. **动力学计算**：除了运动学，Robotics TOOLBOX还处理机器人的动力学问题，包括惯性参数计算、力矩传递分析以及动力学方程的建立，这对于理解机器人在各种任务中的行为至关重要。 3. **轨迹生成**：工具箱包含了生成机器人运动轨迹的功能，这对于规划机器人在空间中的平滑移动路径非常重要。 4. **MEX文件支持**：引入了MEX文件，这是一种编译后的MATLAB代码，可以显著提高计算性能，尤其对于复杂的机器人算法来说，这极大地提高了效率。 5. **Simulink支持**：与MATLAB的Simulink集成，使得用户可以直接在Simulink环境中设计和仿真机器人系统的动态行为，这对于系统级的建模和控制策略设计非常有用。 6. **用户社区**：这个工具箱拥有活跃的用户社区，通过邮件列表，用户可以与其他研究者和工程师交流，分享问题和解决方案。 Robotics TOOLBOX的应用场景非常广泛，不仅限于学术研究，也适用于工业自动化、服务机器人以及教育领域。通过这个工具包，用户能够快速原型设计和测试机器人控制系统，无论是理论分析还是实际操作，都能得到强大的支持。随着版本的更新，开发者持续添加新的特性，以满足不断发展的机器人技术需求。 Robotics TOOLBOX for MATLAB是机器人工程和科学研究中不可或缺的工具，它简化了复杂计算，提高了工作效率，促进了机器人技术的创新和应用。

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, but 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

× z

. 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 !

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 "

the angle between the x

i−1

and x

axes about the z

i−1

axis.

For a revolute axis "

is the joint variable and d

is constant, while for a prismatic joint d

is variable, and "

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

−1

(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

exists. 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 !

i−1

not a

and !

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 =

−1

˙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

= f (

) =

(

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

= 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

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

+ 86

25n

+ 66

66,271 51,548

+171

+ 53

n +129

+ 42

−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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