Analytical Mechanics: Link Mechanisms - Beyond hirai/edu/2017/analyticalmechanics/handout/... ...

Click here to load reader

download Analytical Mechanics: Link Mechanisms - Beyond hirai/edu/2017/analyticalmechanics/handout/...  Analytical

of 28

  • date post

    31-Jul-2018
  • Category

    Documents

  • view

    216
  • download

    1

Embed Size (px)

Transcript of Analytical Mechanics: Link Mechanisms - Beyond hirai/edu/2017/analyticalmechanics/handout/... ...

  • Analytical Mechanics: Link Mechanisms

    Shinichi Hirai

    Dept. Robotics, Ritsumeikan Univ.

    Shinichi Hirai (Dept. Robotics, Ritsumeikan Univ.)Analytical Mechanics: Link Mechanisms 1 / 28

  • Agenda

    Agenda

    1 Open Link MechanismKinematics of Open Link MechanismDynamics of Open Link Mechanism

    2 Closed Link MechanismKinematics of Closed Link MechanismDynamics of Closed Link Mechanism

    Shinichi Hirai (Dept. Robotics, Ritsumeikan Univ.)Analytical Mechanics: Link Mechanisms 2 / 28

  • Open Link Mechanism Kinematics of Open Link Mechanism

    Kinematics of two link open mechanism

    1

    2l1

    l2

    link 1

    link 2

    joint 1

    joint 2

    lc2

    lc1

    x

    y

    two link open link mechanismli length of link ilci distance btw. joint i and

    the center of mass of link imi mass of link iJi inertia of moment of link i

    around its center of mass1 rotation angle of joint 12 rotation angle of joint 2

    Shinichi Hirai (Dept. Robotics, Ritsumeikan Univ.)Analytical Mechanics: Link Mechanisms 3 / 28

  • Open Link Mechanism Kinematics of Open Link Mechanism

    Kinematics of two link open mechanism

    position of the center of mass of link 1:

    xc1=

    [xc1yc1

    ]= lc1

    [C1S1

    ]position of the center of mass of link 2:

    xc2=

    [xc2yc2

    ]= l1

    [C1S1

    ]+ lc2

    [C1+2S1+2

    ]orientation angle of link 1:

    1

    orientation angle of link 2:1 + 2

    Shinichi Hirai (Dept. Robotics, Ritsumeikan Univ.)Analytical Mechanics: Link Mechanisms 4 / 28

  • Open Link Mechanism Dynamics of Open Link Mechanism

    Kinetic energy

    velocity of the center of mass of link 1:

    xc1 = lc11

    [S1C1

    ]angular velocity of link 1:

    1

    kinetic energy of link 1:

    T1 =1

    2m1xTc1xc1 +

    1

    2J1

    21

    =1

    2(m1l

    2c1 + J1)

    21

    Shinichi Hirai (Dept. Robotics, Ritsumeikan Univ.)Analytical Mechanics: Link Mechanisms 5 / 28

  • Open Link Mechanism Dynamics of Open Link Mechanism

    Kinetic energy

    velocity of the center of mass of link 2:

    xc2 = l11

    [S1C1

    ]+ lc2(1 + 2)

    [S1+2C1+2

    ]angular velocity of link 2:

    1 + 2

    kinetic energy of link 2:

    T2 =1

    2m2xTc2xc2 +

    1

    2J2(1 + 2)

    2

    =1

    2m2{l21 21 + l2c2(1 + 2)2 + 2l1lc2C21(1 + 2)}+

    1

    2J2(1 + 2)

    2

    Shinichi Hirai (Dept. Robotics, Ritsumeikan Univ.)Analytical Mechanics: Link Mechanisms 6 / 28

  • Open Link Mechanism Dynamics of Open Link Mechanism

    Kinetic energy

    total kinetic energy

    T = T1 + T2 =1

    2

    [1 2

    ] [ H11 H12H21 H22

    ] [12

    ]where

    H11 = J1 +m1l2c1 + J2 +m2(l

    21 + l

    2c2 + 2l1lc2C2)

    H22 = J2 +m2l2c2

    H12 = H21 = J2 +m2(l2c2 + l1lc2C2)

    inertia matrix

    H=

    [H11 H12H21 H22

    ]Shinichi Hirai (Dept. Robotics, Ritsumeikan Univ.)Analytical Mechanics: Link Mechanisms 7 / 28

  • Open Link Mechanism Dynamics of Open Link Mechanism

    Partial derivatives

    H11 and H12 = H21 depend on 2:

    H112

    = 2h12,H122

    =H212

    = h12 (h12= m2l1lc2S2)

    H11 = 2h122, H12 = H21 = h122T

    1= H111 + H122,

    T

    2= H211 + H222

    ddt

    T

    1= H111 H111 H122 H122

    = 2h1212 + h1222 H111 H122

    ddt

    T

    2= H211 H211 H222 H222

    = h1212 H211 H222Shinichi Hirai (Dept. Robotics, Ritsumeikan Univ.)Analytical Mechanics: Link Mechanisms 8 / 28

  • Open Link Mechanism Dynamics of Open Link Mechanism

    Partial derivatives

    H11, H22, and H12 = H21 are independent of 1

    T

    1=

    1

    2

    [1 2

    ] [ 0 00 0

    ] [12

    ]= 0

    H11 and H12 = H21 depend on 2

    T

    2=

    1

    2

    [1 2

    ] [ 2h12 h12h12 0

    ] [12

    ]= h1221 h1212

    contribution of kinetic energy:

    T

    1 d

    dt

    T

    1= 2h1212 + h12

    22 H111 H122

    T

    2 d

    dt

    T

    2= h1221 H211 H222

    Shinichi Hirai (Dept. Robotics, Ritsumeikan Univ.)Analytical Mechanics: Link Mechanisms 9 / 28

  • Open Link Mechanism Dynamics of Open Link Mechanism

    Gravitational potential energy

    potential energy of link 1:

    U1 = m1g yc1 = m1g lc1S1

    potential energy of link 2:

    U2 = m2g yc2 = m2g (l1S1 + lc2S1+2)

    potential energy:U = U1 + U2

    partial derivatives w.r.t. joint angles:

    U

    1= G1 + G12,

    U

    2= G12

    whereG1 = (m1lc1 +m2l1) gC1, G12 = m2lc2 gC1+2

    Shinichi Hirai (Dept. Robotics, Ritsumeikan Univ.)Analytical Mechanics: Link Mechanisms 10 / 28

  • Open Link Mechanism Dynamics of Open Link Mechanism

    Work done by actuator torques

    work done by 1 applied to rotational joint 1:

    11

    work done by 2 applied to rotational joint 2:

    22

    work done by the two actuator torques:

    W = 11 + 22

    partial derivatives w.r.t. joint angles:

    W

    1= 1,

    W

    2= 2

    Shinichi Hirai (Dept. Robotics, Ritsumeikan Univ.)Analytical Mechanics: Link Mechanisms 11 / 28

  • Open Link Mechanism Dynamics of Open Link Mechanism

    Lagrange equations of motion

    Lagrangian:L = T U +W

    Lagrange equations of motion

    L

    1 d

    dt

    L

    1= 0

    L

    2 d

    dt

    L

    2= 0

    let 1= 1 and 2

    = 2:

    H111 H122 + h1222 + 2h1212 G1 G12 + 1 = 0 H222 H121 h1221 G12 + 2 = 0

    Shinichi Hirai (Dept. Robotics, Ritsumeikan Univ.)Analytical Mechanics: Link Mechanisms 12 / 28

  • Open Link Mechanism Dynamics of Open Link Mechanism

    Lagrange equations of motion

    canonical form of ordinary differential equations:[12

    ]=

    [12

    ][H11 H12H21 H22

    ] [12

    ]=

    [h12

    22 + 2h1212 G1 G12 + 1

    h1221 G12 + 2

    ]state variables: joint angles 1, 2 and angular velocities 1, 2

    the inertia matrix is regular 2nd eq. is solvable we can compute 1 and 2

    1, 2, 1, 2 are functions of 1, 2, 1, 2

    we can sketch 1, 2, 1, 2 using any ODE solver(e.g. Runge-Kutta method)

    Shinichi Hirai (Dept. Robotics, Ritsumeikan Univ.)Analytical Mechanics: Link Mechanisms 13 / 28

  • Open Link Mechanism Dynamics of Open Link Mechanism

    PD control

    1 = KP1(d1 1) KD11

    2 = KP2(d2 2) KD22[

    12

    ]=

    [12

    ][H11 H12H21 H22

    ] [12

    ]=

    [ + KP1(d1 1) KD11 + KP2(d2 2) KD22

    ]

    current values of 1, 2, 1, 2

    their time derivatives 1, 2, 1, 2Shinichi Hirai (Dept. Robotics, Ritsumeikan Univ.)Analytical Mechanics: Link Mechanisms 14 / 28

  • Open Link Mechanism Dynamics of Open Link Mechanism

    PI control

    1 = KP1(d1 1) + KI1

    t0

    {(d1 1()} d

    2 = KP2(d2 2) + KI2

    t0

    {(d2 2()} d

    Introduce additional variables:

    1=

    t0

    {(d1 1()} d

    2=

    t0

    {(d2 2()} d

    1 = d1 1, 1 = KP1(d1 1) + KI11

    2 = d2 2, 2 = KP2(d2 2) + KI22

    Shinichi Hirai (Dept. Robotics, Ritsumeikan Univ.)Analytical Mechanics: Link Mechanisms 15 / 28

  • Open Link Mechanism Dynamics of Open Link Mechanism

    PI control

    [12

    ]=

    [12

    ][H11 H12H21 H22

    ] [12

    ]=

    [ + KP1(d1 1) + KI11 + KP2(d2 2) + KI22

    ]1 =

    d1 1

    2 = d2 2

    current values of 1, 2, 1, 2, 1, 2

    their time derivatives 1, 2, 1, 2, 1, 2Shinichi Hirai (Dept. Robotics, Ritsumeikan Univ.)Analytical Mechanics: Link Mechanisms 16 / 28

  • Closed Link Mechanism Kinematics of Closed Link Mechanism

    Kinematics of four-link closed mechanism

    x

    y

    O

    link 1

    link 2

    link 3

    link 4

    1

    3

    24

    tip point

    l1

    l2

    l3

    l4

    (x1,y1) (x3,y3)

    joint 1

    joint 2

    joint 3

    joint 4

    joint 5

    joint 1, 3: activejoint 2, 4, 5: passive

    1, 2, 3, 4:rotation angles

    1, 3:actuator torques

    Shinichi Hirai (Dept. Robotics, Ritsumeikan Univ.)Analytical Mechanics: Link Mechanisms 17 / 28

  • Closed Link Mechanism Kinematics of Closed Link Mechanism

    Kinematics of four-link closed mechanism

    decomposition of closed link mechanism into open link mechanisms:left arm link 1 and 2right arm link 3 and 4

    end point of the left arm:[x1y1

    ]+ l1

    [C1S1

    ]+ l2

    [C1+2S1+2

    ]end point of the right arm:[

    x3y3

    ]+ l3

    [C3C3

    ]+ l4

    [C3+4S3+4

    ]two constraints:

    X= l1C1 + l2C1+2 l3C3 l4C3+4 + x1 x3 = 0

    Y= l1S1 + l2S1+2 l3S3 l4S3+4 + y1 y3 = 0

    Shinichi Hirai (Dept. Robotics, Ritsumeikan Univ.)Analytical Mechanics: Link Mechanisms 18 / 28

  • Closed Link Mechanism Dynamics of Closed Link Mechanism

    Lagrangian

    Lagrangian of the closed link mechanism:

    L = Lleft + Lright + xX + yY ,

    Lleft, Lright Lagrangians of the left and right armsx , y Lagrange multipliers

    Lagrange equations of motion:

    L

    1 d

    dt

    L

    1= 0

    L

    2 d

    dt

    L

    2= 0

    L

    3 d

    dt

    L

    3= 0

    L

    4 d

    dt

    L

    4= 0

    Shinichi Hirai (Dept. Robotics, Ritsumeikan Univ.)Analytical Mechanics: Link Mechanisms 19 / 28

  • Closed Link Mechanism Dynamics of Closed Link Mechanism

    Contributions of Lleft

    contributions of Lagrangian Lleft to the four Lagrange eqs:

    H111 H122 + h1222 + 2h1212 G1 G12 + 1, H222 H211 h1221 G12,0,

    0

    where

    H11 = J1 +m1l2c1 + J2 +m2(l

    21 + l

    2c2 + 2l1lc2C2),

    H22 = J2 +m2l2c2,

    H12 = H21 = J2 +m2(l2c2 + l1lc2C