[<< wikibooks] Robotics Kinematics and Dynamics/Serial Manipulator Dynamics
== Acceleration of a Rigid Body ==
The linear and angular accelerations are the time derivatives of the linear and angular velocity vectors at any instant:

  
    
      
        
          
          
            a
          
        
        
          
            
              v
              ˙
            
          
        
        =
        
          
            
              
                d
                
                  
                  
                    a
                  
                
                v
              
              
                d
                t
              
            
          
        
        =
        
          lim
          
            Δ
            t
            →
            0
          
        
        
          
            
              
                
                  
                  
                    a
                  
                
                v
                (
                t
                +
                Δ
                t
                )
                −
                
                  
                  
                    a
                  
                
                v
                (
                t
                )
              
              
                Δ
                t
              
            
          
        
      
    
    {\displaystyle _{a}{\dot {v}}={\dfrac {d\,_{a}v}{dt}}=\lim _{\Delta t\rightarrow 0}{\dfrac {_{a}v(t+\Delta t)-\,_{a}v(t)}{\Delta t}}}
  ,and:

  
    
      
        
          
          
            a
          
        
        
          
            
              ω
              ˙
            
          
        
        =
        
          
            
              
                d
                
                  
                  
                    a
                  
                
                ω
              
              
                d
                t
              
            
          
        
        =
        
          lim
          
            Δ
            t
            →
            0
          
        
        
          
            
              
                
                  
                  
                    a
                  
                
                ω
                (
                t
                +
                Δ
                t
                )
                −
                
                  
                  
                    a
                  
                
                ω
                (
                t
                )
              
              
                Δ
                t
              
            
          
        
      
    
    {\displaystyle _{a}{\dot {\omega }}={\dfrac {d\,_{a}\omega }{dt}}=\lim _{\Delta t\rightarrow 0}{\dfrac {_{a}\omega (t+\Delta t)-\,_{a}\omega (t)}{\Delta t}}}
  The linear velocity, as seen from a reference frame 
  
    
      
        {
        a
        }
      
    
    {\displaystyle \{a\}}
  , of a vector 
  
    
      
        q
      
    
    {\displaystyle q}
  , relative to frame 
  
    
      
        {
        b
        }
      
    
    {\displaystyle \{b\}}
   of which the origin coincides with 
  
    
      
        {
        a
        }
      
    
    {\displaystyle \{a\}}
  , is given by:

  
    
      
        
          
          
            a
          
        
        
          v
          
            q
          
        
        =
        
          
          
            a
          
          
            b
          
        
        R
        
          
          
            b
          
        
        
          v
          
            q
          
        
        +
        
          
          
            a
          
        
        
          ω
          
            b
          
        
        ×
        
          
          
            a
          
          
            b
          
        
        R
        
          
          
            b
          
        
        q
      
    
    {\displaystyle _{a}v_{q}=\,_{a}^{b}R\,_{b}v_{q}+\,_{a}\omega _{b}\times \,_{a}^{b}R\,_{b}q}
  Differentiating the above expression gives the acceleration of the vector 
  
    
      
        q
      
    
    {\displaystyle q}
  :

  
    
      
        
          
          
            a
          
        
        
          
            
              
                v
                ˙
              
            
          
          
            q
          
        
        =
        
          
            
              d
              
                d
                t
              
            
          
        
        
          
          
            a
          
          
            b
          
        
        R
        
          
          
            b
          
        
        
          v
          
            q
          
        
        +
        
          
          
            a
          
        
        
          
            
              
                ω
                ˙
              
            
          
          
            b
          
        
        ×
        
          
          
            a
          
          
            b
          
        
        R
        
          
          
            b
          
        
        q
        +
        
          
          
            a
          
        
        
          ω
          
            b
          
        
        ×
        
          
            
              d
              
                d
                t
              
            
          
        
        
          
          
            a
          
          
            b
          
        
        R
        
          
          
            b
          
        
        q
      
    
    {\displaystyle _{a}{\dot {v}}_{q}={\dfrac {d}{dt}}\,_{a}^{b}R\,_{b}v_{q}+\,_{a}{\dot {\omega }}_{b}\times \,_{a}^{b}R\,_{b}q+\,_{a}\omega _{b}\times {\dfrac {d}{dt}}\,_{a}^{b}R\,_{b}q}
  The equation for the linear velocity may also be written as:

  
    
      
        
          
          
            a
          
        
        
          v
          
            q
          
        
        =
        
          
            
              d
              
                d
                t
              
            
          
        
        
          
          
            a
          
          
            b
          
        
        R
        
          
          
            b
          
        
        q
        =
        
          
          
            a
          
          
            b
          
        
        R
        
          
          
            b
          
        
        
          v
          
            q
          
        
        +
        
          
          
            a
          
        
        
          ω
          
            b
          
        
        ×
        
          
          
            a
          
          
            b
          
        
        R
        
          
          
            b
          
        
        q
      
    
    {\displaystyle _{a}v_{q}={\dfrac {d}{dt}}\,_{a}^{b}R\,_{b}q=\,_{a}^{b}R\,_{b}v_{q}+\,_{a}\omega _{b}\times \,_{a}^{b}R\,_{b}q}
  Applying this result to the acceleration leads to:

  
    
      
        
          
          
            a
          
        
        
          
            
              
                v
                ˙
              
            
          
          
            q
          
        
        
          =
          
            a
          
          
            b
          
        
        R
        
          
          
            b
          
        
        
          
            
              
                v
                ˙
              
            
          
          
            q
          
        
        +
        
          
          
            a
          
        
        
          ω
          
            b
          
        
        ×
        
          
          
            a
          
          
            b
          
        
        R
        
          
          
            b
          
        
        
          v
          
            q
          
        
        +
        
          
          
            a
          
        
        
          
            
              
                ω
                ˙
              
            
          
          
            b
          
        
        ×
        
          
          
            a
          
          
            b
          
        
        R
        
          
          
            b
          
        
        q
        +
        
          
          
            a
          
        
        
          ω
          
            b
          
        
        ×
        
          (
          
            
              
              
                a
              
              
                b
              
            
            R
            
              
              
                b
              
            
            
              v
              
                q
              
            
            +
            
              
              
                a
              
            
            
              ω
              
                b
              
            
            ×
            
              
              
                a
              
              
                b
              
            
            R
            
              
              
                b
              
            
            q
          
          )
        
      
    
    {\displaystyle _{a}{\dot {v}}_{q}=_{a}^{b}R\,_{b}{\dot {v}}_{q}+\,_{a}\omega _{b}\times \,_{a}^{b}R\,_{b}v_{q}+\,_{a}{\dot {\omega }}_{b}\times \,_{a}^{b}R\,_{b}q+\,_{a}\omega _{b}\times \left(^{b}_{a}R\,_{b}v_{q}+\,_{a}\omega _{b}\times \,_{a}^{b}R\,_{b}q\right)}
  In the case the origins of 
  
    
      
        {
        a
        }
      
    
    {\displaystyle \{a\}}
   and 
  
    
      
        {
        b
        }
      
    
    {\displaystyle \{b\}}
   do not coincide, a term for the linear acceleration of 
  
    
      
        {
        b
        }
      
    
    {\displaystyle \{b\}}
  , with respect to 
  
    
      
        {
        a
        }
      
    
    {\displaystyle \{a\}}
  , is added:

  
    
      
        
          
          
            a
          
        
        
          
            
              
                v
                ˙
              
            
          
          
            q
          
        
        =
        
          
          
            a
          
        
        
          
            
              
                v
                ˙
              
            
          
          
            b
            ,
            o
            r
            g
          
        
        +
        
          
          
            a
          
          
            b
          
        
        R
        
          
          
            b
          
        
        
          
            
              
                v
                ˙
              
            
          
          
            q
          
        
        +
        
          
          
            a
          
        
        
          ω
          
            b
          
        
        ×
        
          
          
            a
          
          
            b
          
        
        R
        
          
          
            b
          
        
        
          v
          
            q
          
        
        +
        
          
          
            a
          
        
        
          
            
              
                ω
                ˙
              
            
          
          
            b
          
        
        ×
        
          
          
            a
          
          
            b
          
        
        R
        
          
          
            b
          
        
        q
        +
        
          
          
            a
          
        
        
          ω
          
            b
          
        
        ×
        
          (
          
            
              
              
                a
              
              
                b
              
            
            R
            
              
              
                b
              
            
            
              v
              
                q
              
            
            +
            
              
              
                a
              
            
            
              ω
              
                b
              
            
            ×
            
              
              
                a
              
              
                b
              
            
            R
            
              
              
                b
              
            
            q
          
          )
        
      
    
    {\displaystyle _{a}{\dot {v}}_{q}=\,_{a}{\dot {v}}_{b,org}+\,_{a}^{b}R\,_{b}{\dot {v}}_{q}+\,_{a}\omega _{b}\times \,_{a}^{b}R\,_{b}v_{q}+\,_{a}{\dot {\omega }}_{b}\times \,_{a}^{b}R\,_{b}q+\,_{a}\omega _{b}\times \left(^{b}_{a}R\,_{b}v_{q}+\,_{a}\omega _{b}\times \,_{a}^{b}R\,_{b}q\right)}
  For rotational joints, 
  
    
      
        
          
          
            b
          
        
        q
      
    
    {\displaystyle _{b}q}
   is constant, and the above expression simplifies to:

  
    
      
        
          
          
            a
          
        
        
          
            
              
                v
                ˙
              
            
          
          
            q
          
        
        =
        
          
          
            a
          
        
        
          
            
              
                v
                ˙
              
            
          
          
            b
            ,
            o
            r
            g
          
        
        +
        
          
          
            a
          
        
        
          
            
              
                ω
                ˙
              
            
          
          
            b
          
        
        ×
        
          
          
            a
          
          
            b
          
        
        R
        
          
          
            b
          
        
        q
        +
        
          
          
            a
          
        
        
          ω
          
            b
          
        
        ×
        
          (
          
            
              
              
                a
              
            
            
              ω
              
                b
              
            
            ×
            
              
              
                a
              
              
                b
              
            
            R
            
              
              
                b
              
            
            q
          
          )
        
      
    
    {\displaystyle _{a}{\dot {v}}_{q}=\,_{a}{\dot {v}}_{b,org}+\,_{a}{\dot {\omega }}_{b}\times \,_{a}^{b}R\,_{b}q+\,_{a}\omega _{b}\times \left(_{a}\omega _{b}\times \,_{a}^{b}R\,_{b}q\right)}
  The angular velocity of a frame 
  
    
      
        {
        c
        }
      
    
    {\displaystyle \{c\}}
  , rotating relative to a frame 
  
    
      
        {
        b
        }
      
    
    {\displaystyle \{b\}}
  , which in itself is rotating relative to the reference frame 
  
    
      
        {
        a
        }
      
    
    {\displaystyle \{a\}}
  , with respect to 
  
    
      
        {
        a
        }
      
    
    {\displaystyle \{a\}}
  , is given by:

  
    
      
        
          
          
            a
          
        
        
          ω
          
            c
          
        
        =
        
          
          
            a
          
        
        
          ω
          
            b
          
        
        +
        
          
          
            a
          
          
            b
          
        
        R
        
          
          
            b
          
        
        
          ω
          
            c
          
        
      
    
    {\displaystyle _{a}\omega _{c}=\,_{a}\omega _{b}+\,_{a}^{b}R\,_{b}\omega _{c}}
  Differentiating leads to:

  
    
      
        
          
          
            a
          
        
        
          
            
              
                ω
                ˙
              
            
          
          
            c
          
        
        =
        
          
          
            a
          
        
        
          
            
              
                ω
                ˙
              
            
          
          
            b
          
        
        +
        
          
            
              d
              
                d
                t
              
            
          
        
        
          
          
            a
          
          
            b
          
        
        R
        
          
          
            b
          
        
        
          ω
          
            c
          
        
      
    
    {\displaystyle _{a}{\dot {\omega }}_{c}=\,_{a}{\dot {\omega }}_{b}+{\dfrac {d}{dt}}\,_{a}^{b}R\,_{b}\omega _{c}}
  Replacing the last term with one of the expressions derived earlier:

  
    
      
        
          
          
            a
          
        
        
          
            
              
                ω
                ˙
              
            
          
          
            c
          
        
        =
        
          
          
            a
          
        
        
          
            
              
                ω
                ˙
              
            
          
          
            b
          
        
        +
        
          
          
            a
          
        
        
          ω
          
            b
          
        
        ×
        
          
          
            a
          
          
            b
          
        
        R
        
          
          
            b
          
        
        
          ω
          
            c
          
        
      
    
    {\displaystyle _{a}{\dot {\omega }}_{c}=\,_{a}{\dot {\omega }}_{b}+\,_{a}\omega _{b}\times \,_{a}^{b}R\,_{b}\omega _{c}}
  


== Inertia Tensor ==
The inertia tensor can be thought of as a generalization of the scalar moment of inertia:

  
    
      
        
          
          
            a
          
        
        I
        =
        
          
            (
            
              
                
                  
                    I
                    
                      x
                      x
                    
                  
                
                
                  −
                  
                    I
                    
                      x
                      y
                    
                  
                
                
                  −
                  
                    I
                    
                      x
                      z
                    
                  
                
              
              
                
                  
                    I
                    
                      x
                      y
                    
                  
                
                
                  
                    I
                    
                      y
                      y
                    
                  
                
                
                  −
                  
                    I
                    
                      y
                      z
                    
                  
                
              
              
                
                  
                    I
                    
                      x
                      z
                    
                  
                
                
                  −
                  
                    I
                    
                      y
                      z
                    
                  
                
                
                  
                    I
                    
                      z
                      z
                    
                  
                
              
            
            )
          
        
      
    
    {\displaystyle _{a}I={\begin{pmatrix}I_{xx}&-I_{xy}&-I_{xz}\\I_{xy}&I_{yy}&-I_{yz}\\I_{xz}&-I_{yz}&I_{zz}\\\end{pmatrix}}}
  


== Newton's and Euler's equation ==
The force 
  
    
      
        F
      
    
    {\displaystyle F}
  , acting at the center of mass of a rigid body with total mass
  
    
      
        m
      
    
    {\displaystyle m}
  , causing an acceleration 
  
    
      
        
          
            
              
                v
                ˙
              
            
          
          
            c
            o
            m
          
        
      
    
    {\displaystyle {\dot {v}}_{com}}
  , equals:

  
    
      
        F
        =
        m
        
          
            
              
                v
                ˙
              
            
          
          
            c
            o
            m
          
        
      
    
    {\displaystyle F=m{\dot {v}}_{com}}
  In a similar way, the moment 
  
    
      
        N
      
    
    {\displaystyle N}
  , causing an angular acceleration 
  
    
      
        
          
            
              ω
              ˙
            
          
        
      
    
    {\displaystyle {\dot {\omega }}}
  , is given by:

  
    
      
        N
        =
        
          
          
            c
          
        
        I
        
          
            
              ω
              ˙
            
          
        
        +
        ω
        ×
        
          
          
            c
          
        
        I
        ω
      
    
    {\displaystyle N=\,_{c}I{\dot {\omega }}+\omega \times \,_{c}I\omega }
  ,where 
  
    
      
        
          
          
            c
          
        
        I
      
    
    {\displaystyle _{c}I}
   is the inertia tensor, expressed in a frame 
  
    
      
        {
        c
        }
      
    
    {\displaystyle \{c\}}
   of which the origin coincides with the center of mass of the rigid body.