Angular Kinematics

Angular kinematics description

• study of motion of an object about a circular path exclusive of the influences of mass and force

  • including angular displacement, velocity and acceleration

• angular motion occurs when all points on an object move in a circular path around the same axis

angular displacement

• angle that it moves through from its stating point relative to an axis of rotation

• theta is used to represent angular displacement

• SI units: radians

• vector variables

• absolute angular position is the position of an object relative to a fixed reference

  • angle between initial and final positions

• relative angular position is the position of an object relative to a plane or line that is capable of moving

relative angles

• relative angle is angle formed between two limb segments

• contention is the in the anatomical position

  • all relative joint angles are at 0º

absolute angles

• absolute angle is measured from an external frame of reference

  • ex. from the vertical or horizontal

What is a radian

• 1 rad: the angle displace at the centre of a circle by an arc equal in length to the circle’s radius

• θ = I/r

  • θ is the angel

  • I is arc length

  • r is radius

• magnitude in radians of one complete radiation (360º) is the length of the entire circumference divided by the radius

  • 2π/r or 2π

converting from degrees to radians

• ? radians = [] degrees x (π radians / 180 degrees)

  • multiply number of degrees by π and divide by 180

angular kinematics

• angualr displacement, velocity and acceleration are vector quantities

  • magnitude and direction

• positive or negative as defined by right hand rule

  • clockwise → negative

  • counterclockwise → positive

• though of as a curved line

  • direction of the turn of the curved line is the direction of the vector

relationship between angular and linear displacement

• angular displace of all points along a rotating object is the same regardless of how large the radius of the circle is

• angular displacement of an object measured anywhere along the length of the object is always the same

  • r is points radius of rotation

  • θ is angular displacement through which rotating body moves

• actual linear path taken for each point depends on distance that point is from radius of rotation

• ℓ_A has greater magnitude than ℓ_B because it is at a greater distance from the radius of rotation

  • ℓ = rθ

Angular velocity

• represented by omega

• angular idpalcenmt that occurs during a given period of time

• vector variable

• SI units: rads/s

Angular to linear velocity

• for a given ω, linear velocities are different depending on the distance from the radius of rotation

  • angular velocities of points A and B are the same

  • linear velocities are different

  • the linear velocity of a given point travelling along an angular path is know as tangential velocity

tangential velocity

V_t=rω

• although tangential velocity occurs as a result of angular motion

  • form of linear velocity

• direction of this velocity is a straight line that is tangent to the circular path

Angular acceleration

• represented by letter alpha α

• change in angular velocity occurring over a given period of time

• vector variable

• SI units: rad/s²

relationship of angular and linear acceleration

• angular acceleration occurs when something starts, stops, speeds up or slows down in a circular path

• component of linear acceleration tangent to the circular path of a point on a rotating object is called the tangential acceleration

  • a_T= ar

• this shows that the raids is still important in determining the effective linear kinematic quantity

• notice that r is common factor that defines the linear representation of the naguaslr displacement velocity and acceleration

• manipulating the radius or the length can affect he linear displacement of velocity

  • common factor in change angular → linear and linear → angular

Centripetal acceleration

• if something is turning around a circle then it required an external force or acceleration to cause it to change direction

  • Newtons 1st law

• this acceleration is called centripetal acceleration

• force applied directed toward the centre of the circle its is a centre seeking force

• equation: a_r= ω²r

• in order ot maintain na object travelling at an angular velocity (ω) in a circular path at a certain radial distance ® an inward accleration of a_r is required

• also referred to as radial acceleration

total acceleration

• when an object is rotting there are two types of linear acceleration acting on the object

• where a_t is tangential acceleration and a_r is radial acceleration

• although tangential and radial acceleration occur as a result of angular motion

  • both are linear accelerations that rely on angular values

• from tangential and radial acceleration a resultant vector can be found known as total acceleration

  • a_total²= a_r² + a_t²

angular kinematics measurement

• in order ot understand how we measure angular movement it is necessary to understand the Criolis effect

  • state that when a mass moves in a specific direction with a velocity (v) and an external angular rate is applied the Coriolis effect generates a force that causes the mass to move perpendicularly

  • the value of this displacement is directly related to the angular rate applied

• two masses oscillating in opposite direction at a constant frequency

• when angular rate is applied the Coriolis effect produced by each mass is in opposite directions

  • moves the masses away from one another which can result in a change in capacitance between the masses

    • capacitance is ability of system to store and electric change

  • measuring this change in capacitance the angular rate can be calculated

Gyroscope of IMU

• constants of a proof mass of four parts that is maintained in continuous oscillating movment

• simultaneously move inward and outward in the horizontal plane

• when we begin to rotate the structure the Coriolis force acting on the moving proof mass causes the vibration to change

• 3 modes depending on the axis along which the angular rotation is applied

• when corollas effect is detected constant motion of the deriving mass will cause change in capacitance that is detected by sensing structure and converted into a voltage signal

  • roll mode - angular rate applied along X-axis M1 and M3 will move up and down out of the plane due to the Coriolis effect

    • causes a change in the roll angle

  • pitch mode - angular rate is applied along Y-axis, M2 and M4 will move up and down out of the plane

    • chases change in pitch angle

  • Yaw mode - angular rate applied along Z-axis, M2 and M4 will move horizontally in opposite directions

    • causes a change in the yaw angle

Wheelchair racing metrics of wheel-mounted IMU

• gyroscope data

  • angular velocity in degrees/s

  • how to convert in m/s or km/hr

  • convert degs/s to rad/s

  • convert rad/s to m/s

  • convert m/s to km/hr

• WC velocity data can be used to calculate

  • force/velocity profile

  • push profile metrics