1. Overview

In mechanical systems, motion refers to the change in the position or orientation of a component or mechanism over time. Motion can be classified according to its direction, path, axis of rotation, repetition pattern, and degree of freedom.

A motion trajectory is the path followed by a reference point of a moving object through space. Analyzing the motion trajectory helps engineers determine the appropriate guide structure, drive system, power transmission mechanism, sensors, and control method for a mechanical system.

Motion trajectories can generally be classified as follows:

• Linear motion

• Rotational motion

• Arc and circular motion

• Reciprocating motion

• Vibratory and oscillating motion

• Helical motion

• Curved motion

• Intermittent and indexing motion

• Compound planar motion

• Spatial motion

A motion trajectory does not merely describe the visible shape of a movement. It is a fundamental design condition that affects load direction, speed variation, inertia, friction, vibration, positioning accuracy, and service life.

2. Distinction Between Motion Trajectory and Motion Characteristics

When analyzing motion, trajectory, displacement, velocity, and acceleration must be distinguished from one another.

2.1 Trajectory

A trajectory is the path formed by an object as it moves through space.

For example, a slider moving horizontally forms a straight-line trajectory, while the end point of a rotating arm forms an arc-shaped trajectory.

2.2 Displacement

Displacement is the change in position from a reference point to the current position.

In linear motion, displacement is generally expressed in millimeters or meters. In rotational motion, it is expressed as an angle or a number of revolutions.

2.3 Velocity

Velocity is the change in position per unit time.

The velocity of linear motion can be expressed as follows:

v = dx / dt

The symbols represent:

• v: Linear velocity

• x: Position or displacement

• t: Time

• dx / dt: Rate of change in position with respect to time

For rotational motion, angular velocity is used instead of linear velocity.

ω = dθ / dt

The symbols represent:

• ω: Angular velocity

• θ: Angular displacement

• t: Time

• dθ / dt: Rate of change in angular displacement with respect to time

2.4 Acceleration

Acceleration is the change in velocity per unit time.

a = dv / dt

The symbols represent:

• a: Acceleration

• v: Velocity

• t: Time

• dv / dt: Rate of change in velocity with respect to time

As acceleration increases, the inertial force acting on the mechanical structure also increases.

F = m × a

The symbols represent:

• F: Force acting on the object

• m: Mass of the moving object

• a: Acceleration

Therefore, even when the trajectory is the same, the required motor capacity, frame rigidity, bearing life, and vibration characteristics may vary depending on the acceleration and deceleration conditions.

3. Linear Motion

3.1 Definition

Linear motion is motion in which the reference point of an object moves along a straight line.

It is one of the most common forms of motion in mechanical systems and is used for transferring, pressing, cutting, feeding, discharging, and lifting operations.

3.2 Main Characteristics

• The direction of movement remains constant.

• Position and travel distance can be controlled relatively easily.

• The direction of movement can be clearly constrained by a guide structure.

• High repeatability can be achieved.

• It can be applied to long travel distances.

3.3 Typical Mechanisms

Linear motion can be implemented using the following mechanisms:

• Linear guide and ball screw

• Linear guide and timing belt

• Rack and pinion

• Chain and sprocket

• Pneumatic cylinder

• Hydraulic cylinder

• Linear actuator

• Linear motor

• Cam and slider

• Crank and slider

3.4 Types of Linear Motion

Horizontal Linear Motion

Horizontal linear motion occurs when an object moves in a horizontal direction.

It is used in conveyors, transfer tables, packaging machines, cutting machines, and pick-and-place systems.

Vertical Linear Motion

Vertical linear motion occurs when an object moves parallel to the direction of gravity.

It is used in lifting systems, elevators, presses, and mold opening and closing mechanisms.

In vertical motion systems, the following measures should be considered to prevent uncontrolled falling due to the weight of the moving component:

• Brake motor

• Back-driving prevention mechanism

• Counterweight

• Gas spring

• Safety nut

• Fall-prevention device

• Dual-support structure

Inclined Linear Motion

Inclined linear motion follows a straight path at an angle relative to the horizontal or vertical direction.

It is used in inclined conveyors, lifting feeders, and product discharge chutes.

As the inclination angle increases, gravitational sliding and reverse-direction loads also increase.

3.5 Design Considerations

• Travel distance

• Maximum speed

• Acceleration and deceleration time

• Moving mass

• External load

• Guide spacing

• Moment load

• Positioning accuracy

• Repeatability

• Backlash

• Lubrication and contamination conditions

• Horizontal or vertical installation

In linear-motion systems, the drive function and guide function are generally separated. A ball screw or belt transmits the driving force, while a linear guide supports the load and moment.

4. Rotational Motion

4.1 Definition

Rotational motion occurs when an object changes its angular position around a fixed or moving axis of rotation.

Rotational motion is the basic output form of an electric motor. It can be converted into other forms of motion through gears, pulleys, sprockets, cams, and cranks.

4.2 Main Characteristics

• Continuous motion can be generated.

• High-speed operation is relatively easy to achieve.

• Power transmission efficiency is generally high.

• Torque and speed can be adjusted through a reduction ratio.

• Rotational motion can be readily converted into linear or reciprocating motion.

4.3 Typical Components

• Electric motor

• Gear reducer

• Gear

• Pulley and belt

• Sprocket and chain

• Shaft

• Coupling

• Bearing

• Rotary table

• Rotary actuator

4.4 Types of Rotational Motion

Continuous Rotation

The shaft continuously rotates in one direction.

This motion is used in fans, pumps, mixers, conveyor drive shafts, cutting blades, and screw feeders.

Bidirectional Rotation

The shaft alternately rotates clockwise and counterclockwise.

This motion is used in valve drives, door-opening mechanisms, rotary clamps, and robot joints.

Limited-Angle Rotation

The rotating component moves only within a specified angular range.

This motion is used in rotary cylinders, dampers, flaps, and rotary grippers.

Intermittent Rotation

The rotating component moves through a specified angle, stops, and then rotates again.

This motion is used in indexing tables, turrets, packaging machines, and assembly systems.

4.5 Rotational Inertia

In rotational motion, both the mass of the rotating object and the distance of that mass from the axis of rotation are important.

Rotational inertia can be expressed as follows:

J = Σ(m × r²)

The symbols represent:

• J: Mass moment of inertia

• Σ: Sum of all individual mass elements

• m: Mass

• r: Distance from the axis of rotation

• r²: Square of the rotational radius

As the rotational radius increases, the moment of inertia increases significantly. Therefore, more torque is required for acceleration and deceleration.

4.6 Design Considerations

• Rotational speed

• Required torque

• Rotational inertia

• Shaft-support method

• Bearing arrangement

• Shaft torsion

• Eccentricity and imbalance

• Critical speed

• Backlash

• Gearbox efficiency

• Stopping-position accuracy

• Frequency of forward and reverse operation

5. Arc and Circular Motion

5.1 Definition

Arc motion occurs when an object moves along part of a circular path around a fixed center of rotation.

Circular motion occurs when an object moves along a complete circular path.

Rotational motion generally describes the angular change of the object itself, whereas arc and circular motion describe the path formed by a specific point on the moving object.

5.2 Typical Structures

• Rotating arm

• Link arm

• Rotary table

• Turret

• Swing gate

• Rotary gripper

• Crank mechanism

• Rotary feeding device

5.3 Characteristics

• The farther a point is from the axis of rotation, the higher its linear speed.

• Large inertial forces may occur at the end of a rotating arm.

• Directional changes can be achieved within a limited space.

• An object can be moved without using a linear guide.

• Structural components must be arranged around the center of rotation.

The linear velocity at the end of a rotating arm can be calculated as follows:

v = r × ω

The symbols represent:

• v: Linear velocity at the end of the arm

• r: Radius from the center of rotation to the end point

• ω: Angular velocity

Even at the same angular velocity, the linear velocity increases as the radius increases.

5.4 Applications

• Robot arms

• Door hinges

• Rotary covers in food-processing machinery

• Product-orientation mechanisms

• Rotary cutting blades

• Rotary filling machines

• Circular indexing tables

5.5 Design Considerations

• Rotational radius

• Working range

• Interference zone

• Shaft and bearing loads

• Rotating-arm rigidity

• End-point inertia

• Safety-guard coverage

• Braking distance during emergency stops

6. Reciprocating Motion

6.1 Definition

Reciprocating motion is the repeated back-and-forth movement of an object between two positions.

Although the trajectory itself is usually linear or arc-shaped, reciprocating motion is distinguished from unidirectional motion because the direction of movement reverses periodically.

6.2 Typical Mechanisms

• Slider-crank mechanism

• Eccentric cam

• Pneumatic cylinder

• Hydraulic cylinder

• Ball-screw actuator

• Rack and pinion

• Scotch-yoke mechanism

• Reciprocating cam

• Belt-driven slider

6.3 Characteristics

• A specified travel range is repeated.

• Velocity decreases at both end positions before the direction reverses.

• High acceleration may occur during reversal.

• Fatigue life becomes important when the number of cycles is high.

• Impact and vibration can occur easily.

6.4 Applications

• Presses

• Reciprocating pumps

• Reciprocating compressors

• Cutting devices

• Sealing devices

• Pushers

• Product-alignment mechanisms

• Slicing machines

• Sealing bars in packaging machinery

6.5 Motion Characteristics

In a slider-crank mechanism, the slider velocity is not constant even when the crank rotates at a constant speed.

The slider reaches its highest velocity near the middle of the stroke and its velocity becomes zero at both ends of the stroke.

This nonlinear motion directly affects production speed and impact load.

6.6 Design Considerations

• Stroke length

• Number of reciprocating cycles per minute

• Impact during direction reversal

• Fatigue of links and pins

• End-of-stroke cushioning

• Vibration and noise

• Repeatability

• Lubrication conditions

• Position during emergency stopping

In high-speed reciprocating systems, the mass of the moving components should be minimized, and a low-jerk velocity profile should be applied.

7. Vibratory and Oscillating Motion

7.1 Definition

Vibratory motion is repeated movement over a short distance or small angular range around an equilibrium position.

Oscillating motion generally refers to repeated angular movement from side to side within a limited range.

7.2 Vibratory Motion

Vibration may occur in either a linear or rotational direction.

Typical vibration-generating methods include:

• Eccentric rotating mass

• Electromagnetic vibrator

• Piezoelectric actuator

• Cam mechanism

• Crank mechanism

• Pneumatic vibrator

7.3 Oscillating Motion

Oscillating motion is repeated rotational movement within a specified angular range around an axis.

Typical mechanisms include:

• Rocker arm

• Four-bar linkage

• Crank-rocker mechanism

• Rotary cylinder

• Cam follower

• Servo-motor forward and reverse drive

7.4 Applications

• Vibratory feeders

• Screening equipment

• Product-alignment devices

• Mixing equipment

• Vibratory chutes in packaging machines

• Wiper mechanisms

• Oscillating nozzles

• Oscillating washing systems

7.5 Design Considerations

• Frequency

• Amplitude

• Natural frequency

• Resonance

• Damping

• Vibration transmitted to the frame

• Loosening of fasteners

• Repeated fatigue

• Noise

• Potential product damage

Resonance may occur when the operating frequency approaches the natural frequency of the structure.

Resonance can produce a large amplitude from a relatively small input. It should therefore be avoided unless it is intentionally used as part of the system function.

8. Helical Motion

8.1 Definition

Helical motion combines rotational motion and axial linear motion.

The reference point of the object rotates around an axis while simultaneously moving along that axis, thereby forming a helical path.

8.2 Typical Structures

• Lead screw

• Ball screw

• Threaded joint

• Screw conveyor

• Extrusion screw

• Drill

• Auger

• Helical cam

8.3 Characteristics

• Rotational motion can be converted into linear motion.

• A large axial force can be generated from a relatively small rotational input.

• Travel distance and speed are determined by the screw lead.

• Self-locking may be achieved through thread friction.

• A geometric relationship exists between rotation and axial displacement.

The axial travel of a screw can be calculated as follows:

x = n × L

The symbols represent:

• x: Axial travel distance

• n: Number of screw revolutions

• L: Screw lead

For example, when a screw with a lead of 10 mm rotates five times:

x = 5 × 10 mm = 50 mm

8.4 Applications

• Ball-screw transfer systems

• Screw presses

• Product-extrusion equipment

• Screw feeders

• Height-adjustment mechanisms

• Screw-fastening systems

• Drilling machines

• Product-transfer systems in meat-processing machinery

8.5 Design Considerations

• Screw lead

• Rotational speed

• Axial load

• Buckling

• Critical rotational speed

• Screw efficiency

• Backlash

• Lubrication

• Wear

• Cleanability of food-contact surfaces

• Back-driving capability

9. Curved Motion

9.1 Definition

Curved motion occurs when an object follows a continuous path that is neither a straight line nor a simple circular arc.

A curved trajectory may be generated using a curved guide rail, cam profile, linkage mechanism, or multi-axis control system.

9.2 Implementation Methods

• Curved guide rail

• Cam and follower

• Linkage mechanism

• Simultaneous X-axis and Y-axis control

• Robot manipulator

• Curved belt conveyor

• Profile rail

9.3 Characteristics

• Complex working paths can be generated.

• Structural obstacles can be avoided.

• Working direction and orientation can be changed continuously.

• Velocity and acceleration conditions vary according to curvature.

• Control and analysis are more complex than for linear motion.

The centripetal acceleration in curved motion can be calculated as follows:

a_c = v² / r

The symbols represent:

• a_c: Centripetal acceleration

• v: Velocity along the curved path

• r: Radius of curvature

• v²: Square of the velocity

At the same velocity, centripetal acceleration increases as the radius of curvature decreases.

When the velocity doubles, the centripetal acceleration becomes four times greater. Therefore, speed and lateral load must be evaluated more carefully in curved sections than in straight sections.

9.4 Applications

• Curved conveyors

• Automated welding paths

• Cutting paths

• Robotic pick-and-place systems

• Product-orientation mechanisms

• Cam-driven packaging machines

• Machining equipment for complex shapes

9.5 Design Considerations

• Minimum radius of curvature

• Velocity variation by section

• Lateral load on the guide

• Follower contact pressure

• Trajectory error

• Structural interference

• Cable and hose bending

• Lubrication and wear

10. Intermittent and Indexing Motion

10.1 Definition

Intermittent motion converts continuous input motion into alternating periods of movement and dwell.

Indexing motion moves an object by a predetermined distance or angle and then stops it accurately at a specified position.

10.2 Typical Mechanisms

• Geneva mechanism

• Cam indexer

• Barrel cam

• Ratchet and pawl

• Escapement mechanism

• Servo indexing table

• Pneumatic stopper

• Clutch and brake

10.3 Characteristics

• Movement and dwell are repeated.

• Sufficient dwell time can be secured at the working position.

• Multiple processes can be arranged sequentially.

• Stopping-position accuracy is important.

• Impact may occur during starting and stopping.

10.4 Applications

• Rotary indexing tables

• Automatic assembly machines

• Filling machines

• Labeling equipment

• Packaging machinery

• Turret mechanisms

• Product-distribution systems

• Multi-process machining equipment

10.5 Design Considerations

• Index angle

• Dwell time

• Movement time

• Stopping-position accuracy

• Dwell rigidity

• Impact and vibration

• Maximum allowable inertia

• Number of indexing cycles per minute

• Position retention during an emergency stop

• Sensor and homing method

11. Compound Planar Motion

11.1 Definition

Compound planar motion occurs when an object simultaneously performs linear and rotational movement in the same plane, or when two or more axis motions are combined to form a complex trajectory.

11.2 Typical Mechanisms

• Four-bar linkage

• Slider-crank mechanism

• Scotch-yoke mechanism

• Planar cam

• Pantograph

• XY table

• Planar robot

• Planar-motion section of a Delta or SCARA mechanism

11.3 Characteristics

• The relationship between input and output motion may be nonlinear.

• The trajectory is determined by link length and joint position.

• A mechanically defined trajectory can be generated repeatedly.

• Multi-axis control allows the trajectory to be changed.

• Clearances between links may accumulate.

11.4 Applications

• Linkage systems in packaging machines

• Product-transfer arms

• Folding and unfolding mechanisms

• Wiper mechanisms

• Robotic handling systems

• Multi-axis cutting machines

• Automatic feeding devices

11.5 Design Considerations

• Degree of freedom

• Singularities

• Link interference

• Joint clearance

• Trajectory accuracy

• Input-shaft torque

• Output-speed variation

• Fatigue life of links

• Assembly tolerances

12. Spatial Motion

12.1 Definition

Spatial motion occurs when an object changes both its position and orientation in three-dimensional space.

The position of an object can be expressed using X, Y, and Z coordinates. Its orientation can be expressed using rotation angles or roll, pitch, and yaw.

12.2 Degrees of Freedom

A rigid body can have up to six degrees of freedom in three-dimensional space:

• Linear movement along the X-axis

• Linear movement along the Y-axis

• Linear movement along the Z-axis

• Rotation around the X-axis

• Rotation around the Y-axis

• Rotation around the Z-axis

12.3 Typical Systems

• Six-axis articulated robot

• Delta robot

• SCARA robot

• Parallel robot

• Stewart platform

• Five-axis machine tool

• Automated handling system

• Robotic gripper system

12.4 Characteristics

• Complex positions and orientations can be achieved.

• The system can respond to various working directions.

• Trajectory planning and control are complex.

• Motion at one joint can influence other joints.

• Singularities and collision zones must be evaluated.

12.5 Design Considerations

• Working envelope

• Degrees of freedom

• Payload

• End-effector velocity

• Repeatability

• Joint rigidity

• Singularities

• Self-collision

• Interference with surrounding structures

• Cable and hose management

• Safety zones

13. Follower Motion

13.1 Definition

Follower motion occurs when a mechanical element moves along a predetermined profile or guide path.

Typical examples include a cam follower moving along a cam profile and a roller moving along a shaped guide rail.

13.2 Implementation Methods

• Cam and follower

• Grooved cam

• Track cam

• Profile guide

• Copying mechanism

• Mechanical template

• Curved rail

13.3 Characteristics

• A specified motion law can be implemented mechanically.

• Repeated motion can be generated without a complex control system.

• Displacement, velocity, and acceleration are determined by the cam profile.

• High surface pressure may occur at the contact point.

• The machining accuracy of the profile determines motion accuracy.

13.4 Design Considerations

• Cam profile

• Pressure angle

• Follower size

• Contact stress

• Lubrication

• Wear

• Follower separation

• Spring preload

• Continuity of velocity and acceleration

• Noise

14. Relationship Between Motion Trajectory and Drive Method

The same motion trajectory can be produced using several different drive methods.

For example, linear motion can be produced by a pneumatic cylinder, ball screw, belt, rack and pinion, or linear motor.

The drive method should be selected according to the following criteria.

14.1 Pneumatic Drive

Suitable conditions:

• Simple reciprocating motion

• Fast operation

• Relatively low positioning accuracy

• Clean working environment

• Simplified mechanical structure

Limitations:

• Precise control is difficult because air is compressible.

• Air consumption may be high.

• Speed may vary depending on the load.

14.2 Hydraulic Drive

Suitable conditions:

• Large thrust

• Heavy loads

• Impact loads

• Low-speed, high-force operation

Limitations:

• Risk of oil leakage

• Contamination-control requirements

• Higher maintenance burden

• Additional hygienic measures required for food-processing machinery

14.3 Electric Drive

Suitable conditions:

• Precise positioning

• Velocity and acceleration control

• Multi-axis synchronization

• Applications requiring programmable changes

Typical drive sources:

• Servo motor

• Stepper motor

• AC motor

• DC motor

• Linear motor

14.4 Mechanical Drive

Suitable conditions:

• Highly repeatable fixed trajectories

• High-speed continuous production

• Simplified control system

• Mechanical synchronization of multiple motions

Typical elements:

• Cam

• Linkage

• Crank

• Gear

• Geneva mechanism

15. Velocity Profiles in Trajectory Design

Even when the motion trajectory is identical, the operating quality of the machine varies depending on the velocity profile.

15.1 Trapezoidal Velocity Profile

A trapezoidal velocity profile consists of three stages:

• Acceleration

• Constant velocity

• Deceleration

It is simple to implement, but sudden changes in acceleration can cause impact and vibration.

15.2 Triangular Velocity Profile

A triangular velocity profile is used when the travel distance is too short for the system to reach a constant-velocity stage.

The system changes directly from acceleration to deceleration.

15.3 S-Curve Velocity Profile

An S-curve velocity profile limits jerk, which is the rate of change in acceleration.

Jerk can be expressed as follows:

Jerk = Change in acceleration / Change in time

It can also be written as:

j = da / dt

The symbols represent:

• j: Jerk

• a: Acceleration

• t: Time

• da / dt: Rate of change in acceleration with respect to time

An S-curve velocity profile provides the following advantages:

• Reduced impact

• Reduced vibration

• Reduced product movement or sloshing

• Longer machine service life

• Improved high-speed positioning performance

It is suitable for food-processing, packaging, semiconductor, and precision-assembly equipment in which product stability and positioning are important.

16. Criteria for Selecting a Motion Trajectory

A motion trajectory should not be selected solely according to the required function of the machine. Space, speed, load, accuracy, hygiene, safety, and maintainability must also be considered.

16.1 Functional Conditions

• What is being moved?

• From where to where must it be moved?

• Must its orientation change during movement?

• Is the operation continuous or intermittent?

• Is a process performed while the object is stationary?

16.2 Load Conditions

• Moving mass

• External working load

• Impact load

• Moment load

• Direction of gravity

• Eccentric load

16.3 Performance Conditions

• Travel distance

• Maximum velocity

• Acceleration

• Cycle time

• Positioning accuracy

• Repeatability

• Dwell time

16.4 Spatial Conditions

• Installation area

• Height restrictions

• Rotational radius

• Structural interference

• Operator access space

• Maintenance space

16.5 Environmental Conditions

• Water and cleaning agents

• Dust

• High and low temperatures

• Corrosive environment

• Possibility of food contact

• Ingress-protection requirements

16.6 Safety Conditions

• Pinch hazards

• Cutting hazards

• Collision hazards

• Falling hazards

• Exposed rotating components

• Residual movement after an emergency stop

• Position retention during a power failure

17. Comparison of Typical Applications by Motion Trajectory

18. Design Verification Items

After selecting the motion trajectory, the following items should be verified.

18.1 Kinematic Verification

• Ability to reach the target position

• Overall trajectory

• Velocity and acceleration

• Link angles

• Singularities

• Motion range

18.2 Dynamic Verification

• Drive torque

• Inertial force

• Impact load

• Bearing reaction force

• Shaft load

• Frame vibration

18.3 Structural Verification

• Deformation

• Stress

• Buckling

• Fatigue

• Resonance

• Fastener strength

18.4 Interference Verification

• Collision between components

• Interference between the mechanism and the workpiece

• Interference with safety guards

• Cable and hose interference

• Operator accessibility

18.5 Service-Life Verification

• Guide life

• Bearing life

• Ball-screw life

• Belt fatigue

• Chain wear

• Cam-contact fatigue

• Link-pin wear

19. Summary of Main Equations

Linear Velocity

v = dx / dt

Velocity is the rate of change in position with respect to time.

Angular Velocity

ω = dθ / dt

Angular velocity is the rate of change in angular displacement with respect to time.

Acceleration

a = dv / dt

Acceleration is the rate of change in velocity with respect to time.

Force

F = m × a

Force is calculated by multiplying mass by acceleration.

Rotational Inertia

J = Σ(m × r²)

Rotational inertia is calculated by summing the products of each mass and the square of its distance from the axis of rotation.

Linear Velocity in Arc Motion

v = r × ω

The linear velocity at the end point increases as either the radius or angular velocity increases.

Axial Travel of a Screw

x = n × L

Travel distance is calculated by multiplying the number of revolutions by the screw lead.

Centripetal Acceleration in Curved Motion

a_c = v² / r

Centripetal acceleration increases as velocity increases or the radius of curvature decreases.

Jerk

j = da / dt

Jerk is the rate of change in acceleration with respect to time.

20. Conclusion

Motion trajectory is a fundamental design element that converts the intended function of a machine into actual movement.

Linear, rotational, arc, reciprocating, vibratory, helical, curved, intermittent, and spatial motions each have different structural characteristics and loading conditions. Therefore, motion should not be selected only according to the desired path.

The following factors must be considered together:

• Travel path

• Required velocity

• Acceleration and deceleration

• Load and inertia

• Positioning accuracy

• Number of repetitions

• Installation space

• Safety

• Hygiene

• Maintainability

• Drive and control method

Good motion design is not limited to moving an object to a target position. The system must perform the required movement within the specified time and allowable tolerance, without excessive impact or vibration, while maintaining safe and stable operation.

Motion trajectory is not a secondary condition to be considered after the machine structure has been determined. It is the starting point for determining the mechanical structure, drive system, and control method.