Speakers with dysarthria often struggle to accurately pronounce words and effectively communicate with others. Automatic speech recognition (ASR) is a powerful tool for extracting the content from speakers with dysarthria. However, the narrow concept of ASR typically only covers technologies that process acoustic modality signals. In this paper, we broaden the scope of this concept that the generalized concept of ASR for dysarthric speech. Our survey discussed the systems encompassed acoustic modality processing, articulatory movements processing and audio-visual modality fusion processing in the application of recognizing dysarthric speech. Contrary to previous surveys on dysarthric speech recognition, we have conducted a systematic review of the advancements in this field. In particular, we introduced state-of-the-art technologies to supplement the survey of recent research during the era of multi-modality fusion in dysarthric speech recognition. Our survey found that audio-visual fusion technologies perform better than traditional ASR technologies in the task of dysarthric speech recognition. However, training audio-visual fusion models requires more computing resources, and the available data corpus for dysarthric speech is limited. Despite these challenges, state-of-the-art technologies show promising potential for further improving the accuracy of dysarthric speech recognition in the future.