Homogeneous Transforms Convenient representation of rigid body transformations 2-D motion is x|co(△e)cos(△6)△x‖x y|=cos(△)cos(△)y‖y 0 Rotations and translations are now multiplications only 2-D Space: Visibility Graphs qgoal This figure shows the visibility graph G of a simple configuratie clude the edges of CB. The bold line is the shortest path in also in which CB consists of three disconnected c/(Cfree) between qinit and qgoal Image adapted from J C. Latomb What happens in 3-D C-space?Homogeneous Transforms ● Convenient representation of rigid body transformations ● 2-D motion is ● Rotations and translations are now multiplications only ⎥ ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎢ ⎣ ⎡ ⎥ ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎢ ⎣ ⎡ ∆ ∆ ∆ ∆ ∆ ∆ = ⎥ ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎢ ⎣ ⎡ 0 0 1 1 cos( ) cos( ) cos( ) cos( ) 1 ' ' y x y x y x θ θ θ θ 2-D Space: Visibility Graphs What happens in 3-D C-space? This figure shows the visibility graph G of a simple configuration space in which CB consists of three disconnected regions. The links of G also include the edges of CB. The bold line is the shortest path in cl(Cfree) between qinit and qgoal . qinit qgoal Image adapted from J. C. Latombe