Kinematic Differential Equations

Week-8

1Electropaedia

Attitude Kinematics

• In the previous sections, we developed several different ways to describe the attitude, or orientation, of one reference frame with respect to another, in terms of attitude variables.

• In this section, we treat kinematics in which the relativeorientation between two reference frames is time dependent.

• The time-dependent relationship between two referenceframes is described by so called kinematic differentialequations.

• In this section we derive the kinematic differential equationsfor the:– Direction cosine matrix,

– Euler angles, and

– Quaternions.

2

Direction Cosine Matrix• A and B are two reference frames

moving relative to each other.

• The angular velocity vector of reference frame B with respect toreference frame A is denoted by:

• Where the angular velocity vectorω is time dependent.

• We have defined the direction

cosine matrix (DCM)

332211

/

ˆˆˆ bbb

AB

ω

ωω

B/ACC

3

2

1

3

2

1

1

3

2

1

3

2

1

3

2

1

b

b

b

b

b

b

a

a

a

a

a

a

b

b

bT

CCC

3

a1

a3

a2

b1

b2

b3

ω1

ω3

ω2

A

B

• Because the two reference frames are rotating relative toeach other, the direction cosine matrix and its elements Cij

are functions of time.

• Taking the time derivative in A and denoting it by an overdot, we obtain:

Direction Cosine Matrix

3

2

1

12

13

23

3

2

1

3

2

1

3

2

1

3

2

1

3

2

1

0

0

0

0

0

0

0

0

0

b

b

b

b

b

b

b

b

b

b

b

b

b

b

b

b

b

b

TT

TTTT

CC

ω

ω

ω

CCCC

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• Where

• By defining the skew-symmetric matrix:

333231

232221

131211

CCC

CCC

CCC

C

0

0

0

12

13

23

Ω

Direction Cosine Matrix

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• We obtain:

• We obtain:

• Taking transpose of this equation and

0

0

0

3

2

1

b

b

bTT

ΩCC

0 ΩCCTT

ΩΩT

0 CΩC

Direction Cosine Matrix

6

CΩC

• Which is called the kinematic differential equations for the direction cosine matrix C.

• Differential equaitons for each element of C can be written as:

23113233

22112232

21111231

13333123

12332122

11331121

33223313

32222312

31221311

CCC

CCC

CCC

CCC

CCC

CCC

CCC

CCC

CCC

Direction Cosine Matrix

7

Derivation: dcm.m

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• If ω1, ω2, ω3 are known as functions of time, then the orientation of B relative to A as a function of time can be determined by solving these kinematic differential equations.

• These equations can be solved using numerical techniques.

• Orthonormality condition is often used tocheck the accuracy of numerical integration.

• C CT=I

• Example 1: ex1.m

Direction Cosine Matrix

Euler Angles• Like the kinematic differential equation for the direction

cosine matrix C, the orientation of a reference frame B relative to a reference frame A can also be described byintroducing the time dependence of Euler angles.

• Consider the rotational sequence of 3(θ3)-2(θ2)-1(θ1) to Bfrom A. Derivation 1:

• The inverse relationship

9

3

2

1

211

211

2

3

2211211

1

3

2

1

coscossin0

cossincos0

sin01

0

0

)()(

0

0

)(

0

0

CCC

3

2

1

11

2121

21212

2

3

2

1

0

0cos

1

cs

cscc

scssc

ωSθ

θSω

1

• If ω1, ω2, ω3 are known as functions of time, and have initial conditions for the three Euler angles then the orientation of Brelative to A as a function of time can be determined by solving these kinematic differential equations.

• These equations can be solved using numerical techniques.

• However these calculations involve the computation of thetrigonometric functions of the angle.

• Singularites exist when θ2=π/2.

• Such a mathematical singularity problem can be avoided by selecting a different set of Euler angles.

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Euler Angles

• Similarly for 3(Φ)-1(θ)-3(ψ) sequence:

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Euler Angles

0

0

)()(

0

0)(0

0

133

3

2

1

CCC

3

2

1

0

0

sin

1

scccs

sssc

cs

Example 2: example2.mExample 3:example3.m

Rotation Vector Kinematics

• Desired kinematic equation for the rotation vector :

• Inverse is:

12

q

qυ 4

1cos2 q

Quaternions

13

4

3

2

1

4321

3412

2143

1234

3

2

1

2

0 q

q

q

q

qqqq

qqqq

qqqq

qqqq

4

3

2

1

321

312

213

123

4

3

2

1

0

0

0

0

2

1

q

q

q

q

q

q

q

q

Example 4:exeeuler_quat.m:

• In strapdown inertial reference system of aerospace vehicles, the body rates ω1, ω2, ω3 are measured by rate gyros that are «strapped down» to the vehicles.

• The kinematic differential equations is then integrated numerically using onboard flight computer to determine the orientation of the vehicles in terms of quaternions.

• Quaternions have no singularity as do Euler angles.

• Moreover, quaternions are well suited for onboard real-time computation because only products and no trigonometric relations exist in the kinematic differential equations.

• Thus, S/C orientation is now commonly described in terms of quatenions.

In short form:

qω

qωωq

T

2

1

2

1

4

4

q

q

Rodrigues Parameter Kinematics

• The kinematic equation satisfied by the Gibbs vector is most easily obtained from the kinematic equation for the quaternion.

• The inverse of this equation is:

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Modified Rodrigues Parameter Kinematics

• The kinematic equation for the modified Rodrigues parameters can also be obtained from the kinematic equation for the quaternion.

• The inverse of the kinematic equation for the MRPs is:

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CASE STUDY

• B frame 1(a(t))-2(b(t))-1(c(t)) rotates with

– w1=sin(t)*exp(-4*t);

– w2=sin(2*t)*exp(-4*t);

– w3=sin(3*t)*exp(-4*t);

• Relative to A inertial frame.

• If initial values are a(0)=1, b(0)=2, c(0)=3

• Calculate a(5), b(5), c(5)

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