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1. Introduction

1.1. Spatial Mechanics

The principles of spatial mechanics are used to design and analyze mechanical systems in a wide range of applications, including aerospace, automotive, biomedical, and manufacturing industries. To study the kinematics of a mechanical system, it is necessary to understand the geometric relationships between its parts, regardless of the forces acting upon them. This requires the use of advanced mathematical tools, such as vector calculus and matrix algebra, to represent the motion and interactions of the system in a precise and concise manner.

One of the key goals of this course is to develop a mathematical model of a mechanical system that accurately predicts its behavior over time, based solely on the geometry of its motion. By understanding the motion of a system, we can design mechanisms that are efficient and reliable, and optimize their performance for a given set of operating conditions. The principles of spatial mechanics are essential for the design and analysis of a wide range of mechanical systems, and are an important part of many engineering disciplines.

Planar mechanisms and spatial mechanisms are types of mechanical systems that differ in the number of dimensions in which they can move. Planar mechanisms are mechanical systems that are restricted to motion in a single plane, and therefore have two degrees of freedom. Examples of planar mechanisms include four-bar linkage systems, slider-crank mechanisms, and crank-slider mechanisms.

Spatial mechanisms, on the other hand, are mechanical systems that can move in three dimensions and therefore have three degrees of freedom. Examples of spatial mechanisms include robotic arms, aerospace vehicles, and mechanical linkages used in manufacturing and assembly systems.

Some common spatial mechanisms used in industry include:

Stewart Platform: This mechanism consists of a hexagonal base and a platform that is connected to the base by six revolute joints. It allows for three-dimensional motion by adjusting the angles of the joints, and is often used in simulators for pilot training or vehicle driving.

Robotic Arm: A robotic arm is a spatial mechanism that consists of a series of links connected by joints, and is commonly used in manufacturing for pick-and-place tasks. A 4-axis or 6-axis robot arm allows for a wide range of motion and flexibility in positioning items.

Wobble Plate Pumps: These pumps use a spatial mechanism to generate flow and pressure by oscillating a plate or piston back and forth. They are used in a variety of applications, including fluid transfer and fluid power systems.

Gimbaled Mounts: Gimbaled mounts are spatial mechanisms that allow for the orientation of a sensor or camera to be changed with respect to a reference frame. They consist of two or more axes of rotation mounted on a gimbal, which allows the sensor or camera to be pointed in different directions. They are commonly used in satellite and aircraft applications to stabilize and orient sensors and cameras.

1.2. Kinematics, Dynamics, Mechanism, and Mechanics

Kinematics and dynamics are two branches of classical mechanics that study the motion of objects. Kinematics is the study of motion without considering the forces that produce the motion. It is concerned with the position, displacement, rotation, speed, velocity, and acceleration of an object. Dynamics, on the other hand, is the study of the forces that act on an object and their effect on the object’s motion.

A mechanism is a device made up of several interconnected parts that work together to perform a specific task. It typically consists of a series of resistant bodies connected by moving joints, forming a closed kinematic chain with at least one fixed link. Mechanisms are used to transform motion in various ways, such as changing the direction, speed, or magnitude of motion. Mechanics, on the other hand, is the scientific study of the behavior of physical bodies when subjected to forces or displacements, and the effects of these bodies on their environment.

Kinematic analysis and kinematic synthesis are two related methods used to study the motion of mechanisms. Kinematic analysis involves investigating a given mechanism based on its geometry and other known characteristics, such as input angular velocity and angular acceleration. Kinematic synthesis, on the other hand, is the process of designing a mechanism to perform a specific task. It involves determining the geometry and other characteristics of a mechanism that will allow it to perform a desired motion.

1.3. Degrees of Freedom

Three degrees of freedom (3 DOF) refer to the ability of a system to move in three independent directions or to rotate about three independent axes. This allows for a wide range of motion, including translation, rotation, and tilting. In mechanical systems, 3 DOF is often desired in order to provide flexibility and adaptability to different operating conditions and environments.

There are several types of joints that allow for 3 DOF. Some examples include:

Spherical joints: also known as ball joints, these allow for translation in three perpendicular directions and rotation about three perpendicular axes (x,y,z). This type of joint is often used in the industry to connect two rigid bodies in a way that allows them to rotate freely around a common point. A common example of a spherical joint is a ball and socket joint, which is used in the human shoulder and many other applications.

Universal joints: these allow for rotation about two perpendicular axes and are commonly used to connect two shafts that are not aligned.

Helical joints: these allow for translation in a single direction and rotation about two perpendicular axes, and are often used to transmit torque and motion between two rotating shafts.

Planar joints: these allow for motion in a plane and typically have two degrees of freedom, allowing for translation in two perpendicular directions and rotation about a single axis. A common example of a planar joint is a sliding joint, which is used in many mechanical systems to allow for linear motion.

Gruebler’s criterion is a method used to calculate the degrees of freedom (DOF) of a mechanism, which is defined as the number of independent ways in which the mechanism can move. The DOF of a mechanism can be calculated using the following equation: \(DOF = 3(N - 1) - 2P1 - 1P2 - Fr\)


  1. N is the number of links in the mechanism.

  2. P1 is the number of pairs of elements in the mechanism that constrain the motion to 1 degree of freedom.

  3. P2 is the number of pairs of elements in the mechanism that constrain the motion to 2 degrees of freedom.

  4. Fr is the number of redundant pairs in the mechanism. A redundant pair is a pair of elements that does not contribute to the constraint of the mechanism, but may allow for additional motion that may be useful for certain tasks.

For example, consider a mechanism with 4 links, 3 pairs of elements that constrain the motion to 1 degree of freedom, 1 pair of elements that constrain the motion to 2 degrees of freedom, and 1 redundant pair. The DOF of this mechanism can be calculated as follows: \(DOF = 3(4 - 1) - 2(3) - 1(1) - 1 = 1\)

This means that the mechanism has 1 degree of freedom, which means that it can move in one independent direction or rotate about one independent axis.

In summary, kinematic redundancy refers to the presence of redundant pairs in a mechanism, which do not affect the DOF of the mechanism but may allow for additional motion that may be useful for certain tasks.

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Comments (4)

Dr. Sam

Using the knowledge of spatial mechanics and computer programming introduced in this course, you can go on to perform advanced kinematic analysis of 3D mechanisms.

These skills apply to robotics systems design, research, and development.

Joe Disamalu

The notes are on point. I've gained alot of skill and visualization from the unit.


Does it mean the inverse of a 3x3 matrix same as its transpose

Dr. Sam

In spatial mechanics, the inverse and transpose of a 3x3 matrix are not the same, but they can be related in specific cases. The inverse (A⁻¹) is a matrix that, when multiplied by A, results in the identity matrix (A * A⁻¹ = I). The transpose (Aᵀ) is obtained by interchanging the rows and columns of A.

For 3x3 orthogonal matrices, often used to represent rotations, the transpose is equal to the inverse (A⁻¹ = Aᵀ) because they satisfy Aᵀ * A = I. This property is unique to orthogonal matrices, and for general 3x3 matrices, the inverse and transpose are not the same.