Example: kinematic scheme

Example: kinematic scheme

Dimensions

𝜑

𝜔

𝜑

= 180

𝜔

= 2 𝑟𝑎𝑑 𝑠

Analysis based on:

Graphical method (using polygons and vector equations)

Part I of the mechanism

Graphical method (using polygons and vectora equations)

Part II of the mechanism

Slider is the connecting link

𝜔

𝐴 𝐵

𝐷 𝐶

𝐸

2

4 3

STEP I - Velocity

𝜋_𝑣

𝑑𝑖𝑟. 𝑣

𝐴

𝐵 2

𝑣_𝐵𝐴=𝜔₂ ∙ AB

𝜋_𝑣

𝑑𝑖𝑟. 𝑣

=𝑣

𝐴

𝐵 2

𝑣_𝐵 = 𝑣_𝐴 + 𝑣_𝐵𝐴=𝜔₂ ∙ AB=0,2 ^𝑚

𝑠

𝑣

𝜋_𝑣

𝐵

𝐷 𝐶

4 3

𝑑𝑖𝑟. 𝑣

𝑑𝑖𝑟. 𝑣

𝑣

𝑏

𝑣_𝐶 = 𝑣_𝐵 + 𝑣_𝐶𝐵

𝑣_𝐶 = 𝑣_𝐷 + 𝑣_𝐶𝐷=𝑣_𝐶𝐷

𝜋_𝑣

𝑑𝑖𝑟. 𝑣

𝑑𝑖𝑟. 𝑣

𝑏 𝑐

𝐵

𝐷 𝐶

4 3

𝑣

𝜋_𝑣

𝐵

𝐷 𝐶

4 3

𝑐

𝑣

𝑣

𝑣

𝑏

𝜔

accel.

𝑣

= 𝑣

𝜔

=

𝐶𝐷

=

𝑚 𝑠

0,2 𝑚

=0,3418

𝑠

𝜋_𝑣

𝐵

𝐷 𝐶

4 3

𝑐

𝑣

𝑣

𝑣

𝑏

𝑣

𝜔

𝜔

=

𝐶𝐵

=

𝑚 𝑠

0,2 𝑚

=0,8582

𝑠

For accel.

𝜋_𝑣

𝐵

𝐶 3

𝑐

𝑏

𝐸

𝑣_𝐸 = 𝑣_𝐶 + 𝑣_𝐸𝐶

𝑑𝑖𝑟. 𝑣

𝑑𝑖𝑟. 𝑣

𝑣

𝑣

𝜋_𝑣

𝐵

𝐶 3

𝑐

𝑏

𝐸

𝑑𝑖𝑟. 𝑣

𝑑𝑖𝑟. 𝑣

𝑒

𝑣

𝑣

𝜋_𝑣

𝐵

𝐶 3

𝑐

𝑣

𝑣

𝑏

𝐸

𝑒

𝑣

𝑣

𝑣

𝐹

𝑑𝑖𝑟. 𝑣

𝐴

6

𝜋_𝑣

𝑑𝑖𝑟. 𝑣

𝑑𝑖𝑟. 𝑣

𝐹

𝐴 𝐸

𝑐

𝑣

𝑣

𝑏 𝑒

𝑣

𝑣_𝐹 = 𝑣_𝐸 + 𝑣_𝐹𝐸 𝑣_𝐹 = 𝑣_𝐴 + 𝑣_𝐹𝐴=𝑣_𝐹𝐴

5

6 3

𝜋_𝑣

𝑐

𝑣

𝑣

𝑏 𝑒

𝑣

𝑑𝑖𝑟. 𝑣

𝑑𝑖𝑟. 𝑣

𝜋_𝑣

𝑐

𝑣

𝑣

𝑏 𝑒

𝑣

𝑓

𝑣

For accel.

𝑣

𝑣_𝐹𝐸 = 0,17164 𝑚 𝑠

𝑣

= 𝑣

𝜔

𝜔

=

𝐹𝐴

=

𝑚 𝑠

0,35 𝑚

=0,97142

𝑠

For accel.

6

𝜋_𝑣

𝑐

𝑣

𝑣

𝑏 𝑒

𝑣

𝑓

𝑣

𝐴

𝐹 𝐸

𝐵

𝐶

𝐷

𝜋_𝑣

𝑐

𝑣

𝑣

𝑏

𝑒

𝑣

𝑓

𝑣

0,1 𝑚 𝑠 100 [𝑚𝑚]

Velocity scale

polygon v_B 0,2

v_C 0,06836 v_E 0,3246 v_F 0,34

STEP II - Acceleration

𝐴

𝐵 2

𝑎_𝐵 = 𝑣_𝐴 + 𝑎_𝐵𝐴=𝑎_𝐵𝐴^𝑛 + 𝑎_𝐵𝐴^𝑡 = 𝑎_𝐵𝐴^𝑛 = 𝜔₂² ∙ AB=0,4 ^𝑚

𝑠²

𝑎

= 𝑎

0,1 𝑚 𝑠² 100 [𝑚𝑚]

𝜋_𝑎

𝐴

𝐵 2

𝑏

0,1 𝑚 𝑠² 100 [𝑚𝑚]

𝜋_𝑎

𝐵

𝐷 𝐶

4 3

𝑑𝑖𝑟. 𝑎_𝐶𝐵^𝑡

𝑎_𝐶 = 𝑎_𝐵 + 𝑎_𝐶𝐵^𝑛 + 𝑎_𝐶𝐵^𝑡

𝑎_𝐶 = 𝑎_𝐷 + 𝑎_𝐶𝐷^𝑛 + 𝑎_𝐶𝐷^𝑡=𝑎_𝐶𝐷^𝑛 + 𝑎_𝐶𝐷^𝑡 𝑎_𝐶𝐵^𝑛 = 𝜔₃²𝐶𝐵 = 0,1473 ^𝑚

𝑠²

𝑎_𝐶𝐷^𝑛 = 𝜔₄²𝐶𝐷 = 0,02337 ^𝑚

𝑠²

𝑎_𝐶𝐷^𝑛

𝑎_𝐶𝐵^𝑛

𝑎

𝑑𝑖𝑟. 𝑎_𝐶𝐷^𝑡

𝑏

0,1 𝑚 𝑠² 100 [𝑚𝑚]

𝜋_𝑎

𝑑𝑖𝑟. 𝑎_𝐶𝐵^𝑡 𝑑𝑖𝑟. 𝑎_𝐶𝐷^𝑡

𝑎_𝐶𝐷^𝑛

𝑎_𝐶𝐵^𝑛

𝑎

𝑐

𝑏

0,1 𝑚 𝑠² 100 [𝑚𝑚]

𝜋_𝑎

𝑎_𝐶𝐵^𝑡 𝑎_𝐶𝐷^𝑡

𝑎_𝐶𝐷^𝑛

𝑎_𝐶𝐵^𝑛

𝑎

𝑎

𝑐

𝑏

0,1 𝑚 𝑠² 100 [𝑚𝑚]

𝜋_𝑎

𝑑𝑖𝑟. 𝑎_𝐸𝐵^𝑡

𝑎_𝐸 = 𝑎_𝐵 + 𝑎_𝐸𝐵^𝑛 + 𝑎_𝐸𝐵^𝑡 𝑎_𝐸 = 𝑎_𝐶 + 𝑎_𝐸𝐶^𝑛 + 𝑎_𝐸𝐶^𝑡 𝑎_𝐸𝐵^𝑛 = 𝜔₃²𝐵𝐶 = 0,26591 ^𝑚

𝑠²

𝑎_𝐸𝐶^𝑛 = 𝜔₃²𝐸𝐶 = 0,22138 ^𝑚

𝑠²

𝑎_𝐸𝐶^𝑛

𝑎_𝐸𝐵^𝑛

𝑑𝑖𝑟. 𝑎_𝐸𝐶^𝑡

𝐵

𝐶 3

𝐸 𝑎

𝑎

𝐸𝐶 = 0,30058 ^𝑚 𝐸𝐵 = 0,36104 𝑚 𝑐

𝑏

0,1 𝑚 𝑠² 100 [𝑚𝑚]

𝜋_𝑎

𝑑𝑖𝑟. 𝑎_𝐸𝐵^𝑡

𝑎_𝐸𝐶^𝑛

𝑎_𝐸𝐵^𝑛

𝑑𝑖𝑟. 𝑎_𝐸𝐶^𝑡

𝐵

𝐶 3

𝐸 𝑎

𝑎

𝑒

𝑐

𝑏

0,1 𝑚 𝑠² 100 [𝑚𝑚]

𝜋_𝑎

𝑎_𝐸𝐵^𝑡

𝑎_𝐸𝐶^𝑛

𝑎_𝐸𝐵^𝑛

𝑎_𝐸𝐶^𝑡

𝐵

𝐶 3

𝐸 𝑎

𝑎

𝑒

𝑐

𝑏

𝑎

0,1 𝑚 𝑠² 100 [𝑚𝑚]

𝜋_𝑎

𝑒

𝑐

𝑏

𝑑𝑖𝑟. 𝑎_𝐹𝐸^𝑡

𝑎_𝐹 = 𝑎_𝐴 + 𝑎_𝐹𝐴^𝑛 + 𝑎_𝐹𝐴^𝑡 = 𝑎_𝐹𝐴^𝑛 + 𝑎_𝐹𝐴^𝑡 𝑎_𝐹 = 𝑎_𝐸 + 𝑎_𝐹𝐸^𝑐 + 𝑎_𝐹𝐸^𝑡

𝑎_𝐹𝐴^𝑛 = 𝜔₆²𝐹𝐴 = 0,33028 ^𝑚

𝑠²

𝑎_𝐹𝐸^𝑐 = 2𝜔₃𝑣_𝐹𝐸 = 0,20547 ^𝑚

𝑠²

𝑎_𝐹𝐸^𝑐

𝑎_𝐹𝐴^𝑛

𝑑𝑖𝑟. 𝑎_𝐹𝐴^𝑡

𝑣

𝜔

𝑎

𝑎

𝑎

𝐹

𝐴 5 𝐸

6 3

0,1 𝑚 𝑠² 100 [𝑚𝑚]

𝑒

𝑐

𝑏

𝑓

𝜋_𝑎

𝑎_𝐹𝐸^𝑐

𝑎_𝐹𝐴^𝑛

𝑎

𝑎

𝑎

𝑑𝑖𝑟. 𝑎_𝐹𝐸^𝑡

𝑑𝑖𝑟. 𝑎_𝐹𝐴^𝑡

𝐹

𝐴 5 𝐸

6 3

0,1 𝑚 𝑠² 100 [𝑚𝑚]

𝜋_𝑎

𝑒

𝑐

𝑏

𝑎_𝐹𝐴^𝑡

𝑎_𝐹𝐸^𝑡

𝑎_𝐹𝐸^𝑐

𝑎_𝐹𝐴^𝑛

𝑎

𝑎

𝑎

𝑓

𝐹

𝐴 5 𝐸

6 3

0,1 𝑚 𝑠² 100 [𝑚𝑚]

𝜋_𝑎

𝑒

𝑐

𝑎

𝑎

𝑎

𝑎

𝑓

0,1 𝑚 𝑠² 100 [𝑚𝑚]

Acceleration scale

polygon a_B 0,4

a_C 0,187362 a_E 0,232849 a_F 0,356729 𝜋_𝑎

𝑒

𝑐

𝑏

𝑓

𝑎

𝑎

𝑎

𝑎

Comparison after numerical method (SAM)

polygon SAM a_B 0,4 0.40000 a_C 0,187362 0.18736 a_E 0,232849 0.23285 a_F 0,356729 0.35639 polygon SAM

v_B 0,2 0.20000 v_C 0,06836 0.06836 v_E 0,3246 0.32460 v_F 0,34 0.34001

SAM velocity example file

SAM acceleration example file

Analitical method (using vectors and loops)

ശ𝑎 𝑏

ശ𝑐

𝑑ശ

First loop

𝑎-𝑏- Ԧ𝑐- Ԧ Ԧ 𝑑=0

𝜃₂

𝜃₄ 𝜃₃

First loop

ശ𝑎 𝑏

ശ𝑐

𝑑ശ

𝑎 sin 𝜃

− 𝑏 sin 𝜃

+ 𝑐 sin 𝜃

-(y

-y

) = 0

Projections

𝑎 cos 𝜃

− 𝑏 cos 𝜃

+ 𝑐 cos 𝜃

-(x

-x

)= 0

X:

Y:

𝑎-𝑏- Ԧ𝑐- Ԧ Ԧ 𝑑=0

Second loop

ശ𝑎 𝑏

𝑎-𝑏+ Ԧ𝑒- Ԧ Ԧ 𝑓=0

𝑓ശ

𝑟₅

𝜃₆

𝜃₃ − 90⁰ 𝜃₂

𝜃₃

ശ𝑎 𝑏

𝑟₅ 𝑓ശ

Projections Second loop

𝑎 sin 𝜃

− 𝑏 sin 𝜃

+ 𝑟

sin(𝜃

− 90

) − 𝑓 sin 𝜃

= 0 𝑎 cos 𝜃

− 𝑏 cos 𝜃

+ 𝑟

cos(𝜃

− 90

) − 𝑓 cos 𝜃

= 0

X:

Y:

𝑎-𝑏+ Ԧ𝑒- Ԧ Ԧ 𝑓=0

𝜃₆

𝜃₃ − 90⁰ 𝜃₂

𝜃₃ 𝑟₅

ശ𝑎 𝑏

𝑓ശ

𝜃₄ ശ𝑐

𝑑ശ

𝑎 sin 𝜃₂ − 𝑏 sin 𝜃₃ + 𝑐 sin 𝜃₄ -(y_D-y_A) = 0

Projections of all loops

𝑎 cos 𝜃₂ − 𝑏 cos 𝜃₃ + 𝑐 cos 𝜃₄-(x_D-x_A)= 0

X:

Y:

𝑎 sin 𝜃₂ − 𝑏 sin 𝜃₃ + 𝑟₅sin(𝜃₃ − 90^𝑜) − 𝑓 sin 𝜃₆= 0 𝑎 cos 𝜃₂ − 𝑏 cos 𝜃₃ + 𝑟₅ cos(𝜃₃ − 90^𝑜) − 𝑓 cos 𝜃₆= 0

X:

Y:

𝜃₆

𝜃₃ − 90⁰ 𝜃₂

𝜃₃ 𝑟₅

ശ𝑎 𝑏

𝑓ശ

𝜃₄ ശ𝑐

𝑑ശ

X:

Y:

X:

Y:

𝑎 ሶ𝜃₂ cos 𝜃₂ − 𝑏 ሶ𝜃₃ cos 𝜃₃ + 𝑐 ሶ𝜃₄ cos 𝜃₄ = 0

Projections of all loops : velocities

−𝑎 ሶ𝜃₂ sin 𝜃₂ + 𝑏 ሶ𝜃₃ sin 𝜃₃ − 𝑐 ሶ𝜃₄ sin 𝜃₄= 0

𝑎 ሶ𝜃₂ cos 𝜃₂ − 𝑏 ሶ𝜃₃ cos 𝜃₃ + 𝑟₅𝜃₃ሶ cos(𝜃₃ − 90^𝑜) + ሶ𝑟₅ sin(𝜃₃ − 90^𝑜) − 𝑓 ሶ𝜃₆ cos 𝜃₆= 0

−𝑎 ሶ𝜃₂ sin 𝜃₂ + 𝑏 ሶ𝜃₃ sin 𝜃₃ − 𝑟₅𝜃₃ሶ sin(𝜃₃ − 90^𝑜) + ሶ𝑟₅ cos(𝜃₃ − 90^𝑜) + 𝑓 ሶ𝜃₆ sin 𝜃₆= 0

sin(𝜃₃ − 90^𝑜) = sin 𝜃₃ cos(90^𝑜) − cos 𝜃₃ sin 90^𝑜 = − cos 𝜃₃ cos(𝜃₃ − 90^𝑜) = cos 𝜃₃ cos(90^𝑜) + sin 𝜃₃ sin 90^𝑜 = sin 𝜃₃

𝑎 ሶ𝜃₂ cos 𝜃₂ − 𝑏 ሶ𝜃₃ cos 𝜃₃ + 𝑟₅𝜃₃ሶ sin 𝜃₃ − ሶ𝑟₅ cos 𝜃₃ − 𝑓 ሶ𝜃₆ cos 𝜃₆= 0

−𝑎 ሶ𝜃₂ sin 𝜃₂ + 𝑏 ሶ𝜃₃ sin 𝜃₃ + 𝑟₅𝜃₃ሶ cos 𝜃₃ + ሶ𝑟₅sin 𝜃₃ + 𝑓 ሶ𝜃₆ sin 𝜃₆= 0

𝜃₆

𝜃₃ − 90⁰ 𝜃₂

𝜃₃ 𝑟₅

ശ𝑎 𝑏

𝑓ശ

𝜃₄ ശ𝑐

𝑑ശ

X:

Y:

X:

Y:

𝑎 ሶ𝜃₂ cos 𝜃₂ − 𝑏 ሶ𝜃₃ cos 𝜃₃ + 𝑐 ሶ𝜃₄ cos 𝜃₄ = 0

Projections of all loops : velocities – matrix form

−𝑎 ሶ𝜃₂ sin 𝜃₂ + 𝑏 ሶ𝜃₃ sin 𝜃₃ − 𝑐 ሶ𝜃₄ sin 𝜃₄= 0

−𝑎 sin 𝜃₂ 𝑎 cos 𝜃₂

−𝑎 sin 𝜃₂ 𝑎 cos 𝜃₂

ሶ𝜃₂ +

𝑏 sin 𝜃₃

−𝑏 cos 𝜃₃

𝑏 sin 𝜃₃ + 𝑟₅cos 𝜃₃

−𝑏 cos 𝜃₃ + 𝑟₅sin 𝜃₃

−𝑐 sin 𝜃₄ 𝑐 cos 𝜃₂

0 0

0 0 sin 𝜃₃

− cos 𝜃₃

0 0 𝑓 sin 𝜃₆

−𝑓 cos 𝜃₆ ሶ𝜃₃ ሶ𝜃₄

ሶ

𝑟₅ ሶ𝜃₆

= 0

𝑎 ሶ𝜃₂cos 𝜃₂ − 𝑏 ሶ𝜃₃ cos 𝜃₃ + 𝑟₅𝜃₃ሶ sin 𝜃₃ − ሶ𝑟₅ cos 𝜃₃ − 𝑓 ሶ𝜃₆ cos 𝜃₆= 0

−𝑎 ሶ𝜃₂ sin 𝜃₂ + 𝑏 ሶ𝜃₃ sin 𝜃₃ + 𝑟₅𝜃₃ሶ cos 𝜃₃ + ሶ𝑟₅ sin 𝜃₃ + 𝑓 ሶ𝜃₆sin 𝜃₆= 0

ሶ𝜃₃ ሶ𝜃₄

ሶ

𝑟₅ ሶ𝜃₆

= −

𝑏 sin 𝜃₃

−𝑏 cos 𝜃₃

𝑏 sin 𝜃₃ + 𝑟₅cos 𝜃₃

−𝑏 cos 𝜃₃ + 𝑟₅sin 𝜃₃

−𝑐 sin 𝜃₄ 𝑐 cos 𝜃₂

0 0

0 0 sin 𝜃₃

− cos 𝜃₃

0 0 𝑓 sin 𝜃₆

−𝑓 cos 𝜃₆

−1 −𝑎 sin 𝜃₂ 𝑎 cos 𝜃₂

−𝑎 sin 𝜃₂ 𝑎 cos 𝜃₂

ሶ𝜃₂

𝑎 ሶ𝜃₂ cos 𝜃₂ − 𝑏 ሶ𝜃₃ cos 𝜃₃ + 𝑐 ሶ𝜃₄ cos 𝜃₄ = 0

Projections of all loops : accelerations

−𝑎 ሶ𝜃₂ sin 𝜃₂ + 𝑏 ሶ𝜃₃ sin 𝜃₃ − 𝑐 ሶ𝜃₄ sin 𝜃₄= 0

𝑎 ሶ𝜃₂ cos 𝜃₂ − 𝑏 ሶ𝜃₃ cos 𝜃₃ + 𝑟₅𝜃₃ሶ sin 𝜃₃ − ሶ𝑟₅ cos 𝜃₃ − 𝑓 ሶ𝜃₆ cos 𝜃₆= 0

−𝑎 ሶ𝜃₂ sin 𝜃₂ + 𝑏 ሶ𝜃₃ sin 𝜃₃ + 𝑟₅𝜃₃ሶ cos 𝜃₃ + ሶ𝑟₅ sin 𝜃₃ + 𝑓 ሶ𝜃₆ sin 𝜃₆= 0

𝑎 ሷ𝜃₂ cos 𝜃₂ − 𝑎 ሶ𝜃₂² sin 𝜃₂ − 𝑏 ሷ𝜃₃ cos 𝜃₃ + 𝑎 ሶ𝜃₃² sin 𝜃₃ + 𝑐 ሷ𝜃₄ cos 𝜃₄ − 𝑐 ሶ𝜃₄² sin 𝜃₄= 0

−𝑎 ሷ𝜃₂ sin 𝜃₂ − 𝑎 ሶ𝜃₂² cos 𝜃₂ + 𝑏 ሷ𝜃₃ sin 𝜃₃ + 𝑏 ሶ𝜃₃² cos 𝜃₃ − 𝑐 ሷ𝜃₄ sin 𝜃₄ − 𝑐 ሶ𝜃₄²cos 𝜃₄= 0

X:

Y:

X:

Y:

X:

Y:

−𝑎 ሷ𝜃₂ sin 𝜃₂ − 𝑎 ሶ𝜃₂² cos 𝜃₂ + 𝑏 ሷ𝜃₃ sin 𝜃₃ + 𝑏 ሶ𝜃₃² cos 𝜃₃ + 𝑟₅𝜃₃ሷ cos 𝜃₃ − 𝑟₅𝜃₃ሶ ² sin 𝜃₃ + + ሶ𝑟₅𝜃₃ሶ c𝑜𝑠 𝜃₃ + ሶ𝑟₅𝜃₃ሶ c𝑜𝑠 𝜃₃ + ሷ𝑟₅ sin 𝜃₃ + 𝑓 ሷ𝜃₆ sin 𝜃₆ + 𝑓 ሶ𝜃₆²cos 𝜃₆= 0

X:

Y:

+ ሶ𝑟₅𝜃₃ሶ sin 𝜃₃ + ሶ𝑟₅𝜃₃ሶ sin 𝜃₃ − ሷ𝑟₅cos 𝜃₃ − 𝑓 ሷ𝜃₆ c𝑜𝑠 𝜃₆ + 𝑓 ሶ𝜃₆² sin 𝜃₆= 0

Projections of all loops : accelerations – matrix form

𝑎 ሷ𝜃₂cos 𝜃₂ − 𝑎 ሶ𝜃₂² sin 𝜃₂ − 𝑏 ሷ𝜃₃ cos 𝜃₃ + 𝑎 ሶ𝜃₃² sin 𝜃₃ + 𝑐 ሷ𝜃₄ cos 𝜃₄ − 𝑐 ሶ𝜃₄²sin 𝜃₄= 0

−𝑎 ሷ𝜃₂sin 𝜃₂ − 𝑎 ሶ𝜃₂² cos 𝜃₂ + 𝑏 ሷ𝜃₃ sin 𝜃₃ + 𝑏 ሶ𝜃₃² cos 𝜃₃ − 𝑐 ሷ𝜃₄ sin 𝜃₄ − 𝑐 ሶ𝜃₄²cos 𝜃₄= 0

X:

Y:

−𝑎 ሷ𝜃₂ sin 𝜃₂ − 𝑎 ሶ𝜃₂² cos 𝜃₂ + 𝑏 ሷ𝜃₃ sin 𝜃₃ + 𝑏 ሶ𝜃₃² cos 𝜃₃ + 𝑟₅𝜃₃ሷ cos 𝜃₃ − 𝑟₅𝜃₃ሶ ² sin 𝜃₃ + + ሶ𝑟₅𝜃₃ሶ c𝑜𝑠 𝜃₃ + ሶ𝑟₅𝜃₃ሶ c𝑜𝑠 𝜃₃ + ሷ𝑟₅ sin 𝜃₃ + 𝑓 ሷ𝜃₆ sin 𝜃₆ + 𝑓 ሶ𝜃₆²cos 𝜃₆= 0

X:

Y:

+ ሶ𝑟₅𝜃₃ሶ sin 𝜃₃ + ሶ𝑟₅𝜃₃ሶ sin 𝜃₃ − ሷ𝑟₅ cos 𝜃₃ − 𝑓 ሷ𝜃₆ c𝑜𝑠 𝜃₆ + 𝑓 ሶ𝜃₆² sin 𝜃₆= 0

−𝑎 sin 𝜃₂ 𝑎 cos 𝜃₂

−𝑎 sin 𝜃₂ 𝑎 cos 𝜃₂

−𝑎 cos 𝜃₂

−𝑎 sin 𝜃₂

−𝑎 cos 𝜃₂

−𝑎 sin 𝜃₂

ሷ𝜃₂ ሶ𝜃₂² +

𝑏 sin 𝜃₃

−𝑏 cos 𝜃₃

𝑏 sin 𝜃₃ + 𝑟₅cos 𝜃₃

−𝑏 cos 𝜃₃ + 𝑟₅sin 𝜃₃

−𝑐 sin 𝜃₄ 𝑐 cos 𝜃₂

0 0

0 0 sin 𝜃₃

− cos 𝜃₃

0 0 𝑓 sin 𝜃₆

−𝑓 cos 𝜃₆ ሷ𝜃₃ ሷ𝜃₄

ሷ

𝑟₅ ሷ𝜃₆

+

𝑏 cos 𝜃₃ 𝑏 sin 𝜃₃

𝑏 cos 𝜃₃ − 𝑟₅sin 𝜃₃ 𝑏 sin 𝜃₃ + 𝑟₅cos 𝜃₃

−𝑐 cos 𝜃₄

−𝑐 sin 𝜃₂ 0 0

0 0 cos 𝜃₃

sin 𝜃₃

0 0 𝑓 cos 𝜃₆

𝑓 sin 𝜃₆

ሶ𝜃₃² ሶ𝜃₄² 2 ሶ𝜃₃𝑟₅ሶ

ሶ𝜃₆²

= 0

Projections of all loops : accelerations – matrix form

−𝑎 sin 𝜃₂ 𝑎 cos 𝜃₂

−𝑎 sin 𝜃₂ 𝑎 cos 𝜃₂

−𝑎 cos 𝜃₂

−𝑎 sin 𝜃₂

−𝑎 cos 𝜃₂

−𝑎 sin 𝜃₂

ሷ𝜃₂ ሶ𝜃₂² +

𝑏 sin 𝜃₃

−𝑏 cos 𝜃₃

𝑏 sin 𝜃₃ + 𝑟₅cos 𝜃₃

−𝑏 cos 𝜃₃ + 𝑟₅sin 𝜃₃

−𝑐 sin 𝜃₄ 𝑐 cos 𝜃₂

0 0

0 0 sin 𝜃₃

− cos 𝜃₃

0 0 𝑓 sin 𝜃₆

−𝑓 cos 𝜃₆ ሷ𝜃₃ ሷ𝜃₄

ሷ

𝑟₅ ሷ𝜃₆

+

𝑏 cos 𝜃₃ 𝑏 sin 𝜃₃

𝑏 cos 𝜃₃ − 𝑟₅sin 𝜃₃ 𝑏 sin 𝜃₃ + 𝑟₅cos 𝜃₃

−𝑐 cos 𝜃₄

−𝑐 sin 𝜃₂ 0 0

0 0 cos 𝜃₃

sin 𝜃₃

0 0 𝑓 cos 𝜃₆

𝑓 sin 𝜃₆

ሶ𝜃₃² ሶ𝜃₄² 2 ሶ𝜃₃𝑟₅ሶ

ሶ𝜃₆²

= 0

ሷ𝜃₃ ሷ𝜃₄

ሷ

𝑟₅ ሷ𝜃₆

= −

𝑏 sin 𝜃₃

−𝑏 cos 𝜃₃

𝑏 sin 𝜃₃ + 𝑟₅cos 𝜃₃

−𝑏 cos 𝜃₃ + 𝑟₅sin 𝜃₃

−𝑐 sin 𝜃₄ 𝑐 cos 𝜃₂

0 0

0 0 sin 𝜃₃

− cos 𝜃₃

0 0 𝑓 sin 𝜃₆

−𝑓 cos 𝜃₆

−1

𝑏 cos 𝜃₃ 𝑏 sin 𝜃₃

𝑏 cos 𝜃₃ − 𝑟₅sin 𝜃₃ 𝑏 sin 𝜃₃ + 𝑟₅cos 𝜃₃

−𝑐 cos 𝜃₄

−𝑐 sin 𝜃₂ 0 0

0 0 cos 𝜃₃ sin 𝜃₃

0 0 𝑓 cos 𝜃₆ 𝑓 sin 𝜃₆

ሶ𝜃₃² ሶ𝜃₄² 2 ሶ𝜃₃𝑟₅ሶ

ሶ𝜃₆² +

+

−𝑎 sin 𝜃₂ 𝑎 cos 𝜃₂

−𝑎 sin 𝜃₂ 𝑎 cos 𝜃₂

−𝑎 cos 𝜃₂

−𝑎 sin 𝜃₂

−𝑎 cos 𝜃₂

−𝑎 sin 𝜃₂

ሷ𝜃₂ ሶ𝜃₂²