ABSTRACT
Spatial mechanisms
A mechanism in which a body moves through a general spatial movement is called a spatial
mechanism. An example is the RSSR linkage, which can be viewed as a four-bar linkage in which
the hinged joints of the coupler link are replaced by rod ends, also called spherical joints or ball
joints. The rod ends allow the input and output cranks of the RSSR linkage to be misaligned to
the point that they lie in different planes, which causes the coupler link to move in a general
spatial movement. Robot arms, Stewart platforms, and humanoid robotic systems are also
examples of spatial mechanisms.
Select this link for an animation of Bennett's linkage, which is a spatial mechanism constructed
from four hinged joints.
In spatial three-degrees-of-freedom (two degrees of translational freedom and one degree of
rotational freedom) parallel manipulator is proposed. The parallel manipulator consists of a
base plate, a movable platform, and three connecting legs. The inverse and forward kinematics
problems are described in closed forms and the velocity equation of the new parallel
manipulator is given. Three kinds of singularities are also presented. The workspace for the
manipulator is analyzed systematically; in particular, indices to evaluate the mobility (in this
paper, mobility means rotational capability) of the moving platform of the manipulator will be
defined and discussed in detail. Finally, a topology architecture of the manipulator is
introduced. The parallel manipulator has wide application in the fields of industrial robots,
simulators, micromanipulators, and parallel machine tools
Chapter 1: Introduction
1.1 Introduction to Spatial Mechanisms
Mechanisms
A machine consists of an actuator input, a system of mechanisms that generate the output
forces and movement, and an interface to the user. Electric
motors, hydraulic and pneumatic actuators provide the input forces and movement. This input
is shaped by mechanisms consisting of gears and gear trains, belt and chain drives, cam and
follower mechanisms, and linkages as well as friction devices such as brakes and clutches.
Structural components consist of the frame, fasteners, bearings, springs, lubricants and seals,
as well as a variety of specialized machine elementssuch as splines, pins and keys.[4][5] The user
interface ranges from switches and buttons to programmable logic controllers and includes the
covers that provide texture, color and styling.
3.1 Planar and Spatial Mechanisms
Mechanisms can be divided into planar mechanisms and spatial mechanisms, according to the
relative motion of the rigid bodies. In a planar mechanisms, all of the relative motions of the
rigid bodies are in one plane or in parallel planes. If there is any relative motion that is not in
the same plane or in parallel planes, the mechanism is called the spatial mechanism. In other
words, planar mechanisms are essentially two dimensional while spatial mechanisms are three
dimensional. This tutorial only covers planar mechanisms.
Spatial mechanisms
A mechanism in which a body moves through a general spatial movement is called a spatial
mechanism. An example is the RSSR linkage, which can be viewed as a four-bar linkage in which the
hinged joints of the coupler link are replaced by rod ends, also called spherical joints or ball joints. The
rod ends allow the input and output cranks of the RSSR linkage to be misaligned to the point that they lie
in different planes, which causes the coupler link to move in a general spatial movement. Robot
arms, Stewart platforms, and humanoid robotic systems are also examples of spatial mechanisms.
Select this link for an animation of Bennett's linkage, which is a spatial mechanism constructed from four
hinged joints.
The group SE(3) is six dimensional, which means the position of a body in space is defined by six
parameters. Three of the parameters define the origin of the moving reference frame relative to the fixed
frame. Three other parameters define the orientation of the moving frame relative to the fixed frame.
Planar mechanism
The majority of mechanisms synthesized and found in application are planar devices. These
mechanisms have motion such that all elements move in one plane or in parallel planes. While
planar mechanisms have been broadly applied, they lack the ability to perform many general
motion-control tasks. In fact, planar mechanisms, as well as spherical mechanisms, form a
subset of spatial mechanisms.
Spatial mechanism
Spatial mechanisms are the most general category of kinematic devices. They offer the greatest
capability to accomplish any desired kinematic task. Spatial mechanisms can be composed of
any number of links and can include joints with any combination of rotational and translational
freed
Planar versus Spatial Mechanisms

A body has planar motion if all of its particles move in parallel planes.

In a spatial mechanism, different particles may move in paths which donʼt all always
remain in a plane.

A body has spherical motion if all of its particles move on the surface of concentric
spheres.

Notice that planar motion is the limiting case of spherical motion when the radius of the
spheres goes to infinity!

For instance, a four-bar linkage is a planar mechanism, even though it is constructed in
three dimensions

This car trunk link mechanism is a rather elegant example of a planar linkage system:

With a planar linkage such as this, you can study the motion by making a twodimensional model, say out of cardboard.

You can project all the points of the different bodies onto one or more reference planes
parallel to the planes traced by the individual particles.

Viewed orthogonally from the side, the parts of the four-bar linkage on the left side of
the car always are aligned with those of the linkage on the right side.

All the hinge axes are perpendicular to a common reference plane which is why the
mechanism remains planar.

The linkage on the right side of the car produces the same motion of the particles as the
one on the left, so from a kinematics point of view it is redundant.Planar versus Spatial
Mechanisms

Viewed orthogonally from theside, the parts of the four-bar linkage on the left side of
the car always are aligned with those of the linkage on the right side.

It can be conceptually “collapsed” into a single planar our-bar linkage model.

Of course, if this were my car, I would notpropose crushing it into a planar
systemcompacted onto that single reference plane.

The added three-dimensional parts add strength and structural stiffness to
themechanism but they donʼt change the kinematics.

It is a planar linkage!
CHAPTER THREE
PATH AND MOTION GENERATION SYNTHESIS OF SPATIAL MECHANISM
Spatial Mechanisms
Spatial linkages can be synthesized for problems such as
• Function generation
• Path generation
• Motion generation
3.1) GENERAL
The path and motion generation synthesis methods which will be explained in this chapter are
the first synthesis methods developed by means of the algebra of exponential rotation
matrices. The main advantage of these methods is that general dyad equations have been
presented for single loop spatial linkages with n links (equations (3.1) and (3.2) ) and very
similar equations can be written for multiloop spatial linkages. Besides in these methods the
mechanism designer directly deals with link lengths and joint angles which make more sense
while in some synthesis methods ,designers work with X ,Y and Z coordinates (ref.18,19,26).
Finally using these synthesis methods designers can benefit the advantages of the algebra of
exponential rotation matrices.
3.2) PATH GENERATION SYNTHESIS
Consider a single loop spatial mechanism with n links. Assume that a coordinate system is
attached to each link according to Denavit-Hartenberg.s convention . Now consider a point P
on the k.th link .
This point -which is called the path tracer point- is supposed to pass through points 0 1 1 , ,..., P
P Pj− which are called precision points.
Figure (3.1) : The vectors which construct the k.th link .
3.4) MOTION GENERATION SYNTHESIS
Consider a single loop spatial linkage with n links. Assume that a coordinate system is attached
to each link according to Denavit-Hartenberg.s convention. Now consider a point P on the k.th
link. A coordinate system (p) is attached to the k.th link so that its origin coincides with the path
tracer point (P) . The coordinate system is supposed to lie in prescribed positions when the path
tracer point passes through the precision points.
Figure (3.5) : k.th link of an n link spatial mechanism.

Even though applications for general spatial mechanisms are less common than for
planar linkages there are many useful spatial devices, such as folding baby strollers and
cribs.

Sometimes things that appear to be planar mechanisms need to be designed for spatial
motions to allow for flexing in the parts.

Some types of aircraft thrust reversers come to mind as an important example:

Wobble plate pumps, bent axis pumps, and swash plate pumps use spatial mechanisms:

simulators for pilot training or for vehicle driving often use a spatial robot called a
“Stewart Platform”.

One of the most common spatial mechanisms is the constant velocity coupling.

Professor Ken Hunt developed a generalized way of designing all theoretically possible
cv joints.
1.2 Mechanism Application
There are a number of reasons for the broad application of planar mechanisms.Well defined
analysis and synthesis techniques are readily available and are taught to many mechanical
engineers. Additionally, there is a variety of synthesis software available for planar mechanism
design, such as LINCAGES, Sphinx, and KINSYN
[Erdman, 1995].
On the other hand, spatial mechanisms have not found wide use and acceptance for several
reasons. One, is that spatial mechanism synthesis and analysis is typically not taught to
engineers in undergraduate education. Techniques for analysis and synthesis of
spatial mechanisms usually involve extensive vector mathematics and linear algebra
techniques. Even for the experienced mechanical designer, it is labor intensive and
difficult to design a spatial mechanism. The visualization of these devices can even be difficult.
Finally, no computer package exists that combines rudimentary training, synthesis, and analysis
of spatial mechanisms in one program.
1.3 Spatial Mechanism Alternatives
Due to the inability of practicing engineers to design spatial mechanisms, many spatial
kinematic tasks are either done by a robotic manipulator, a machine with multiple planar
actuators, or are simply ignored for automation and are performed by human operators.
It should be noted that robotic manipulators provide controlled actuation of all the joints in the
system, which results in a large number of degrees of freedom. This flexibility may be not
necessary for the given task, in which case the manipulator could be replaced with a single
degree-of-freedom spatial mechanism
1.4 Advantages of Spatial Mechanisms

In many automation situations, a spatial mechanism has a clear advantage over robotic
manipulators and combined planar mechanisms.

One good example would be a “pick-and-place” task, the universal task for assembly
machines, that involves noncoplanar motion.

The advantages of spatial, single-degree-of-freedom mechanisms include simplicity,
lower cost, higher reliability, and lower energy consumption.

For these reasons single-degree-of-freedom mechanisms excel over multi-degree-offreedom manipulators in highly repetitive tasks for mass production by reducing startup, maintenance, and operating costs, thereby increasing productivity.

Myklebust, et. al., [1985] expressed it this way: "Spatial mechanisms, being purely
mechanical single-input devices, tend to be more reliable and more energy efficient
than electronically-controlled multiple-input devices such as robotic manipulators. Also,
closed-loop devices, such as spatial mechanisms, are known to be capable of running at
higher speeds and carrying greater loads with more precision than open-loop devices,
such as the typical serial robotic manipulator
1.5 Disadvantages of Spatial Mechanisms

The main disadvantage of spatial mechanisms is the lack of flexibility with respect to
meeting the changing needs of a desired task.

This is a valid point and must be considered before the time and effort is put into
designing a spatial mechanism.

If the mechanism is going to have a short life span and cannot be reused elsewhere,
then it should not be designed and built. On the other hand, if it is going to be an
automotive part, a guidance device on a piece of exercise equipment, or part of a
manufacturingprocess that will not be modified for an extended period of time, then
flexibility becomes less important.

Another potential downside to using spatial mechanisms is the need for custom
fabrication to meet a particular motion requirement. While this may be true, one-off
spatial mechanisms can probably be designed and fabricated for less than the cost of a
robotic manipulator. If the mechanism is going to be mass produced, then the cost and
effort is more easily justified. For instance, consider the top of a convertible car.

It would probably be impractical to employ a multi-degree-of-freedom robotic
manipulator in the back of the car just to open and close the roof. In this case, it is
worth the effort to design a planar mechanism to perform the given task. This same
concept applies to spatial mechanisms.

The last major concern is the difficulty regarding the design of spatial mechanisms.

This problem will persist until a tool is created that will allow the average engineer to
easily synthesize spatial mechanisms. CADSPAM will fulfill this need and
allow designers to exploit the full potential of spatial mechanisms
1.6 Applications for Spatial Mechanisms
If spatial mechanisms could be synthesized quickly and easily, there could be many
suitable
applications
for
them
in
addition
to
the
examples
described
above.
Some examples include the aerospace industry, exercise equipment, and the rehabilitation
medical field. Engineers in aerospace are constantly trying to find ways to make things
compact, light weight, and able to perform a specific spatial function.
For example:
spatial mechanisms are especially useful in satellite design and deployment. The exercise
industry could also use spatial mechanisms to design equipment that more closely mimics the
natural path of the body while performing specific exercises. Human kinetic rehabilitation and
aids for people with disabilities likewise would benefit from such a tool to develop
sophisticated mechanical devices. Basically, spatial mechanisms could be applied to any task
that requires general spatial motion.
Spatial pairs
Prismatic
- 1 DOF
Revolute
- 1 DOF
Spherical
- 3 DOF
Cylindrical - 2 DOF
Helical
- 1 DOF
Planar
-
3 DOF
CHAPTER FOUR
OVER CONSTRAINED SPATIAL MECHANISMS
4.1) GENERAL
The degree of freedom of a spatial mechanism can be calculated by the following formula ,
D.O.F = 6(n −1) −5R −5P − 4C − 3S
(4.1)
Where ,
n is the number of links .
R is the number of revolute joints .
P is the number of prismatic joints .
C is the number of cylindric joints .
S is the number of spheric joints .
When the degree of freedom of a mechanism is less than one , it is expected to be immobile
.However there are some spatial mechanisms whose degrees of freedom - according to formula
(4.1) -are less than unity and still they can move under some specific conditions . Such
mechanisms are called over constrained mechanisms and the conditions under which they
move are called the constraints of the mechanism.
Over constrained mechanisms are attractive to mechanism designers for their higher capacityin comparison to the similar mechanisms-to carry loads and that they are cheaper .A.J.
Shih(ref.30) and J.E.Baker(ref.31,32) recently have worked on some over constrained
mechanisms .In this chapter using the algebra of exponential rotation matrices ,the mobility
and kinematic synthesis of an over constrained spatial mechanism has been studied
4.2) RCCR LINKAGE
According to equation (4.1) the degree of freedom of the RCCR linkage illustrated in Figure (4.1)
is equal to zero but it has been proved that this mechanism is able to move under a constraint .
Figure (4.1) : An RCCR linkage
Figure (4.2) : Schematic figure of the RCCR linkage.
Degree of freedom
The number of independent movements that an object can perform in a 3-D space is called the
number of degrees of freedom (DOF). The movement of an ideal joint is generally associated
with a subgroup of the group of Euclidean displacements. The number of parameters in the
subgroup is called the degrees of freedom (DOF) of the joint.
Degree of freedom is conveniently illustrated for mechanisms with rigid links. The discussion is
limited to mechanisms which obey the general degree-of-freedom equation,
where F = degree of freedom of mechanism,
l = number of links of mechanism,
j = number of joints of mechanism,
fi = degree of freedom of relative motion at ith joint,
σ = summation symbol (summation over all joints), and
λ = mobility number
(the most common cases are λ = 3 for plane mechanisms and λ = 6 for spatial mechanisms).
The degree of freedom of a system can be viewed as the minimum number of coordinates
required to specify a configuration. Applying this definition, we have:
1. For a single particle in a plane two coordinates define its location so it has two degrees
of freedom;
2. A single particle in space requires three coordinates so it has three degrees of freedom;
3. Two particles in space have a combined six degrees of freedom;
4. If two particles in space are constrained to maintain a constant distance from each
other, such as in the case of a diatomic molecule, then the six coordinates must satisfy a
single constraint equation defined by the distance formula. This reduces the degree of
freedom of the system to five, because the distance formula can be used to solve for
the remaining coordinate once the other five are specified.
One-degree-of-freedom (1-Dof)
The position of a single car (engine) moving along a track has one degree of freedom, because
the position of the car is defined by the distance along the track. A train of rigid cars connected
by hinges to an engine still has only one degree of freedom because the positions of the cars
behind the engine are constrained by the shape of the track.
Two-Degree –of-freedom (2-Dof)
A new two-degree-of-freedom (dof) parallel manipulator producing two translations in the
vertical plane. One drawback of existing robots built to realize these dof is their lack of
transversal stiffness, another one being their limited ability to provide very high acceleration.
Indeed, these architectures cannot be lightweight and stiff at the same time. The proposed
parallel architecture is a spatial mechanism which guarantees a good transversal stiffness. It is
composed by two actuated kinematic chains, and two passive chains built in the transversal
plane. The key feature of this robot comes from the two passive chains which are coupled to
create a kinematic constraint: the platform stays in one plane. A stiffness analysis shows that
the robot can be lighter and stiffer than a classical 2-dof mechanism. A prototype of this robot
is presented and preliminary tests show that accelerations above 400 ms−1 can be achieved
while keeping a low tracking error
Three-Degree-of-freedom (3-Dof)
The static balancing of spatial three-degree-of-freedom (3-dof) parallel mechanisms or
manipulators with revolute actuators using counterweights or springs is studied in this paper.
The expressions for the position vector of the center of mass and the total potential energy of
the mechanism are first obtained. Then, the kinematic constraint equations of the mechanism
are introduced in order to eliminate some of the dependent variables from the expressions.
Finally, the conditions for the static balancing of the mechanism are derived from the resulting
expressions. Two examples corresponding to the two balancing methodologies are given in
order to illustrate the results.
CAD model of a spatial 3-dof parallel mechanism with revolute actuators.
Schematic representation of spatial 3-dof parallel mechanism with revolute actuators.
An automobile with highly stiff suspension can be considered to be a rigid body traveling on a
plane (a flat, two-dimensional space). This body has three independent degrees of freedom
consisting of two components of translation and one angle of rotation. Skidding or drifting is a
good example of an automobile's three independent degrees of freedom.
First type of spatial four-dof parallel mechanism with revolute actuators.
Second type of spatial four-dof parallel mechanism with prismatic actuators.
Second type of spatial four-dof parallel mechanism with revolute actuators.
A spatial four-bar mechanism.
The directional view along z-axis for the spatial four-bar mechanism.
.
The DOF of the spatial four-bar mechanism with BC as the output device
six degrees of freedom
The position of a rigid body in space is defined by three components of translation and three
components of rotation, which means that it has six degrees of freedom.
The motion of a ship at sea has the six degrees of freedom of a rigid body, and is described as:
1.Translation:
Translation is the ability to move in three dimensions
1. Moving up and down (heaving);
2. Moving left and right (swaying);
3. Moving forward and backward (surging);
2.Rotation:
Rotation is the ability to change angle around an axis.
1. Tilting forward and backward (pitching);
2. Turning left and right (yawing);
3. Tilting side to side (rolling).
Fig. 1.
Six degrees of freedom.
The fig is six dimensional, which means the position of a body in space is defined by six
parameters. Three of the parameters define the origin of the moving reference frame relative
to the fixed frame. Three of the other parameters define the orientation of the moving frame
relative to the fixed frame.
6DOF compromises translation (movement) in three different axes and rotation in three
different axes. Any object can freely move in 3D space with six degrees of freedom.
We explore and manipulate objects using our hands, moving and rotating them in all directions
and therefore have so called 6 degrees of freedom (6DOF) (see Figure 1). Degrees of freedom
(DOF) describe the motions of an object in 3D space.
This can be broken down into two main groups:
To break down the six degrees of freedom that an object might possess in 3D space, each of the
following is one degree of freedom (Wikipedia,2004):
1. Moving up and down (heaving)
2. Moving left and right (swaying)
3. Moving forward and back (surging)
4. Tilting up and down (pitching)
5. Turning left and right (yawing)
6. Tilting side to side (rolling)
Example : mouse
The conventional WIMP interface uses a 2 DOF mouse. The mouse can only perform two of the
six degrees of freedom. It can move forward and back and left and right. Therefore, the mouse,
as a communication tool to interact with 3D objects for designing, has several serious
shortcomings. As 3D object manipulation and construction requires much more than a 2 DOF
interface.