  Planning the trajectory of the controlled ego vehicle is a key challenge in
automated driving. As for human drivers, predicting the motions of surrounding
vehicles is important to plan the own actions. Recent motion prediction methods
utilize equivariant neural networks to exploit geometric symmetries in the
scene. However, no existing method combines motion prediction and trajectory
planning in a joint step while guaranteeing equivariance under
roto-translations of the input space. We address this gap by proposing a
lightweight equivariant planning model that generates multi-modal joint
predictions for all vehicles and selects one mode as the ego plan. The
equivariant network design improves sample efficiency, guarantees output
stability, and reduces model parameters. We further propose equivariant route
attraction to guide the ego vehicle along a high-level route provided by an
off-the-shelf GPS navigation system. This module creates a momentum from
embedded vehicle positions toward the route in latent space while keeping the
equivariance property. Route attraction enables goal-oriented behavior without
forcing the vehicle to stick to the exact route. We conduct experiments on the
challenging nuScenes dataset to investigate the capability of our planner. The
results show that the planned trajectory is stable under roto-translations of
the input scene which demonstrates the equivariance of our model. Despite using
only a small split of the dataset for training, our method improves L2 distance
at 3 s by 20.6 % and surpasses the state of the art.
