1

Singularly Useful:

Using Kinematic Redundancy to Increase

the Versatility of Industrial Robots Clayton Button

Western Washington University

Manufacturing Engineering Technology

2

Contents

ABSTRACT ....................................................................................................................................................... 3

INTRODUCTION ................................................................................................................................................ 3

MODEL-BASED ROBOT CONTROL ...................................................................................................................... 6

PROPERTIES OF KINEMATIC REDUNDANCY ......................................................................................................... 8

RESOLUTION OF KINEMATIC REDUNDANCY ...................................................................................................... 10

APPLICATIONS OF KINEMATIC REDUNDANCY ................................................................................................... 10

CONCLUSION ................................................................................................................................................. 13

WORKS CITED ............................................................................................................................................... 14

3

ABSTRACT

Demand for versatile industrial robots is increasing, due to changes in the robot

industry’s target demographic. Kinematic redundancy can be exploited to increase the

versatility of manipulators, especially to enable dynamic re-posturing for singularity avoidance

or secondary functionality. The mathematical basis for dynamic re-posturing is briefly

discussed, along with potential and known applications.

INTRODUCTION

This paper begins by broadly summarizing the state of industrial robotics, model-based

control, and modeling kinematic redundancy. The reader who is well-acquainted with these

topics can proceed directly to Resolution of Kinematic Redundancy.

The International Organization for Standardization (ISO) defines an industrial robot as

“an automatically controlled, reprogrammable, multi-purpose manipulator programmable in

three or more axes.” [4] Most industrial robots consist of straight-line linkages, which define

the shape of the robot and bear loads, and joints, the motor-powered and variable connections

of the linkages. Industrial robots can be grouped into three categories: Cartesian (or gantry),

articulated, and parallel. [4] These varieties are differentiated by their joint types; broadly

speaking, joints can be prismatic (linearly actuated) or revolute (rotationally actuated).

Cartesian robots have only prismatic joints, and typically operate in three degrees of freedom.

Articulated robots have revolute joints, although they may also have prismatic joints. An

4

important subtype of articulated robot is the SCARA (Selective Compliance Assembly Robot

Arm); SCARA robots have two revolute joints (with parallel axes of rotation) and a third

prismatic joint. Parallel robots, the least common of the three types, have a fundamentally

different structure than articulated robots: their linkages form closed loops rather than an open

chain, allowing the position and orientation of a platform or end effector to be adjusted by

several independent actuators. Robots that use a combination of parallel and series joints have

been made for research purposes, but are seldom seen in industrial application. [3]

Figure 1

Moving clockwise from the top left – A

Cartesian robot, a six-axis articulated robot,

a SCARA robot, and a parallel “pick-and-

place” robot.

5

While the definition of an industrial robot provides that it must be multi-purpose, some

robots are clearly more versatile than others. A manipulator’s “degrees of freedom” is equal to

the number of variables needed to fully describe its position in three dimensional space. [6] It is

intuitive that the number of applications for a manipulator is proportional to its degrees of

freedom; in other words, the more ways that the manipulator can be moved or contorted, the

greater the number of tasks that can be handled by the manipulator. Unfortunately, increasing

the degrees of freedom necessitates a dramatic increase in the complexity of the robot and its

control, which means a much greater cost. [7] A manipulator that is able to translate its end

effector (the device which interacts with the task environment) in three axes and rotate it in

three axes is considered “fully versatile” for industrial applications. [7] [6] “General purpose

robots” with this ability are being increasingly adopted in modern robotics applications. [2]

Development of industrial robots is primarily guided by two factors: cost and

application. Most of the cost of a robot comes from its “drive line” – the gear boxes, motors,

drive units, rectifiers, etc. that are responsible for powering its movement. Cost strongly

incentivizes the design of manipulators with minimal degrees of freedom for their intended

applications. In past years, the automotive industry was the primary consumer of industrial

robots; in fact, robots were mostly developed for press tending, car body assembly, painting,

and coating. This narrow set of applications led to a convergence of commercial design

concepts, with relatively few exceptions. However, the market for these applications is very

nearly saturated, so industrial robotics will need to expand into new applications quickly in the

coming years. Many of the solutions developed for the auto industry do not translate well into

6

other applications, and industrial robot makers will need to develop new concepts in robot

control. [2]

MODEL-BASED ROBOT CONTROL

Robot control is most generally defined as “All the technologies needed to control the

electromechanical system of an industrial robot.” [2] In the earliest industrial robots, this was

accomplished by repetition of prerecorded changes in joint variables; today, various control

methods expedite or otherwise improve the programming process. “Model-based” robot

control uses computer simulated kinematic and/or dynamic models of the manipulator to

predict motion paths, and allows for “off-line” programming (e.g. programming the robot for a

new application while it is performing its current task), as well as for error-correction routines.

Simulation using dynamic models can be used to estimate control parameters (mathematical

constants used in position calculations) and to optimize such traits as cycle time and

repeatability. Machine learning-based methods have also been explored. [2]

The use of kinematic models allows for the forward or backward calculation of joint

positions from joint variables. The fundamental kinematic mappings are

(1) ( )

(2) ̇ ( ) ̇

where:

q is the vector of joint variables

f is a differentiable nonlinear vector function of q, based on a geometric model of the

manipulator

7

x is the “task vector”, nearly always the location of the end effector, but more generally a direct

kinematic function of joint variables expressed in an a suitable reference frame

J is the Jacobian matrix of the arm, defined as ( )

(the time-derivative of the forward

kinematic equations)

It follows, so long as J(q) is invertible, that

(3) ( )

(4) ̇ ( ) ̇

These are the “inverse kinematic equations”. Of course, J is often singular, so (4) is typically

approximated with a pseudo-inverse of J or numerical methods. [7] [3]

The advantage of this kinematic model-based relationship is that the user can choose the

task vector trajectory ̇ (e.g. the desired velocity of the end effector) and from it, calculate the

necessary rates at which the joint variables must change ( ̇). Because the kinematic equations

provide instantaneous solutions, some method must be used to connect the solutions into

continuous movement. Conventionally, they are iterated many times throughout the

movement in task space, and connected with splining functions. [7]

The drawback of this approach is that certain combinations of ̇ and ̇ result in “kinematic

singularities”. A kinematic singularity refers to a trajectory in task space which cannot be

mapped into joint space; this may stem from several mechanisms. For example, if an industrial

robot is programmed to reach outside of its work envelope (the space composed of possible

positions of the end effector) a singularity will result from the physical limitations of the arm.

Another type of singularity occurs when a task-space trajectory requires an impossibly large

8

velocity in one joint; in the model above, this corresponds to the condition ( ) for

equal-dimensioned task and joint spaces. A spatial example of such a condition is shown in

Figure 2. A third type of singularity results when the mapping from task to joint space is not one

to one; i.e., when there are infinite solutions to (4). This type of singularity results from a

condition known as kinematic redundancy.

PROPERTIES OF KINEMATIC REDUNDANCY

While a general condition for kinematic redundancy may be given in mathematical

terms, a physical description of kinematic redundancy can only be defined with respect to a

particular task. It occurs when a manipulator has enough active degrees of freedom that at

Figure 2

Depiction of a two-dimensional manipulator, with the current position on the right

side of the y-axis, and the desired future position on the left side of the y-axis. A

path for the end effector is specified along decreasing x and y = 0. θ is a joint

variable corresponding to rotation about the origin, and r is a joint variable

corresponding to a telescoping motion of the arm. When r reaches 0, an

instantaneous 180° rotation of θ is required, which corresponds to an angular

velocity of 180/0. [1]

9

least one is arbitrary while performing that task. For example, a commercial six-axis articulated

arc-welding or laser robot will often experience kinematic redundancy; see Figure 3 for an

example.

In the previous example, the kinematic redundancy of the manipulator could be

expressed in terms of a single joint variable. More generally, kinematic redundancy results from

an arbitrary choice in the combination of joint velocities to satisfy (2); this is equivalent to the

columns of ( ) being linearly dependent. As a consequence, the null space of J must have

nontrivial solutions. This is interesting, because it means that ( ) ̇ for some ̇; or

equivalently, the joint variables can be changed in certain ways without movement in task

space. For an industrial robot, this means that the arm can be continually re-postured

Figure 3

This laser-cutting robot’s task trajectory is independent of the value

of the sixth joint variable, making the robot kinematically

redundant (if joint 6 is active) for all laser-cutting tasks.

10

throughout a movement without affecting the trajectory of the end effector. This type of

motion is called “self-movement”. [7]

RESOLUTION OF KINEMATIC REDUNDANCY

The problem posed by kinematic redundancy is that it necessitates an additional set of

heuristics before a joint-space action can be chosen. A simple, but less than ideal, solution to

that problem would be to decrease the number of active joints (e.g. choose not to rotate J6 of

the aforementioned laser-welding robot). This solution is less than ideal because there are

numerous ways to capitalize on the potential re-posturing of the manipulator. For example, an

objective function of q that maps onto null(J) could be optimized. This could mean minimizing

the combined joint actuator power consumption, or maximizing the “manipulability” of the

robot (i.e. orthogonal distance from singularities in joint space).

Another approach would be to increase the dimension of the task space by adding self-

movement constraints, which could be defined based on obstructions in the work envelope or a

cost function associated with self-movement. [7]

APPLICATIONS OF KINEMATIC REDUNDANCY

Some kinematically-redundant manipulator applications have already been identified.

The UC Davis engineering team built a hybrid serial/parallel redundant industrial manipulator

for the application of loading and unloading packages on a flatbed trailer. Their design, dubbed

“UPSarm” (shown in Figure 4), utilizes a closed-loop kinematic chain to provide shoulder

movements, prismatic joints, and a parallel driven wrist subsystem, which together effectively

create ten degrees of freedom in joint space. The design was able to achieve the high strength

11

of closed-kinematic-chain manipulators (a maximum load capacity of 70 lbs) with the versatility

of a serial kinematic manipulator. The kinematic redundancy of the robot allowed it to remain

dexterous throughout its motion in the confines of a flat-bed trailer. [3]

A recent article in IEEE Transactions on Robotics discussed the use of force-motion

control in coordinated robots, dealing with indeterminate environmental geometry. The

authors proposed a theoretical method to allow for a group of industrial robots to synchronize

their movements by sensing the forces exerted by their partners. In order to ease the

complexities of such a complicated system, “The proposed method also exploits redundancy in

robot kinematics in order to keep the overall interconnected system in desired configurations

during its operation.” [5]

Lead through programming is a functionality offered by some robot manufacturers. The

operator activates a mode in which the robot arm goes limp (but balances its weight), and can

be positioned manually. An example of the concept is shown in Figure 4. The positions may

then be recorded via audio commands, or cooperation with a second operator. If a

Figure 4

A schematic diagram and photograph of the UPSarm.

12

kinematically redundant industrial robot were programmed with this method, an objective

function based on maximum manipulability and minimal self-movement could conceivably be

created, allowing the robot to subtly re-posture itself while the operator moves it between

positions, reducing the possibility of encountering singularities during programming or

automatic runs. Alternatively, force sensors on intermediate links of the robot could potentially

enable the operator to reposition the robot while keeping the end effector still.

According to Brogardh:

“…there is a big interest from industry to have robots performing bin picking of

components randomly placed in many layers ... The solution to this problem is of course mainly

given by an intelligent vision system but there are also some tricky problems to solve for the

robot control since all robot movements are random as ordered by the vision system. This

means that the ordered movements may pass or end up in singularities, may need the transition

to a new configuration of the robot arms or the robot wrist...To handle singularities,

functionality for singularity avoidance by tool orientation adjustments can be used and robot

Figure 5

An operator uses both hands to lead an industrial robot

through the necessary motions for an arc-welding task. [2]

13

configuration changes can be handled by automatic analysis of predicted robot configurations

before trajectory generation.”

A kinematically redundant manipulator would have the capability of making such configuration

changes during movement, so long as its joint space trajectory remained in the null space of

J(q).

CONCLUSION

The market for industrial robots is shifting away from the conventional applications in

the automotive business, necessitating new paradigms for development. While manipulators

with high degrees of freedom are more expensive and complicated than those with low degrees

of freedom, they offer greater versatility, which will become increasingly valuable as the

applications for industrial robots continue to diversify. Kinematic redundancy allows for

creative uses of self-movement, which are likely to play an important role in the development

of future control systems. As computational power becomes less expensive and the need for

highly versatile and intuitive manipulators grows, it can be expected that the benefits of

kinematic redundancy will outweigh its costs, and manipulators with many degrees of

kinematic freedom will be developed.

14

WORKS CITED

1. Allen, Peter K., Dr. "CS 4733 Class Notes: Kinematic Singularities and Jacobians."

Columbia University CS 4733. Web. 3 May 2014.

<http://www.cs.columbia.edu/~allen/F13/NOTES/jacobians.pdf>.

2. Brogardh, Tony. "Robot Control Overview: An Industrial Perspective." Modeling,

Identification and Control 30.3 (2009): 167-80. Web. 2 May 2014.

3. Cheng, Harry. "Kinematic Analysis of a Hybrid Serial-and-Parallel Driven Redundant

Industrial Manipulator." International Journal of Robotics & Automation 10.4 (1995):

159-66. Web. 3 May 2014.

4. "Industrial Robots." IFR RSS. International Federation of Robotics, n.d. Web. 04 May

2014. <http://www.ifr.org/industrial-robots/>.

5. Namvar, Mehrzad, and Farhad Aghili. "Adaptive Force-Motion Control of Coordinated

Robots Interacting With Geometrically Unknown Environments." IEEE

TRANSACTIONS ON ROBOTICS 21.4 (2005): n. pag. Web. 3 May 2014.

6. Schilling, Robert J. Fundamentals of Robotics: Analysis and Control. Englewood Cliffs,

NJ: Prentice Hall, 1990. N. pag. Print.

7. Siciliano, Bruno. "Kinematic Control of Redundant Robot Manipulators: A Tutorial."

Journal of Lntelligent and Robotic Systems 212th ser. 3.201 (1990): n. pag. Web. 3 May

2014.